{"id":20120,"date":"2025-12-07T22:09:45","date_gmt":"2025-12-07T16:39:45","guid":{"rendered":"https:\/\/www.geographynotespdf.com\/?p=20120"},"modified":"2026-01-23T09:17:35","modified_gmt":"2026-01-23T03:47:35","slug":"regression-analysis","status":"publish","type":"post","link":"https:\/\/geographynotespdf.com\/cms\/regression-analysis\/","title":{"rendered":"16. Regression Analysis (\u092a\u094d\u0930\u0924\u0940\u0917\u092e\u0928 \u092f\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923)"},"content":{"rendered":"

Regression Analysis <\/strong><\/span><\/h2>\n

(\u092a\u094d\u0930\u0924\u0940\u0917\u092e\u0928 \u092f\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923)<\/strong><\/span><\/p>\n

\n
\n

\"Regression<\/p>\n

\u092a\u0930\u093f\u091a\u092f<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 ‘\u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928’ (Regression) \u0936\u092c\u094d\u0926 \u0915\u093e \u0905\u0930\u094d\u0925 ‘\u092a\u0940\u091b\u0947 \u0932\u094c\u091f\u0928\u093e’ \u0905\u0925\u0935\u093e ‘\u0935\u093e\u092a\u0938 \u0906\u0928\u093e’<\/strong><\/span> (to regress or return back)<\/span><\/strong> \u0939\u0948\u0964 \u0907\u0938 \u0936\u092c\u094d\u0926 \u0915\u093e \u0938\u0930\u094d\u0935\u092a\u094d\u0930\u0925\u092e \u092a\u094d\u0930\u092f\u094b\u0917 1877 \u092e\u0947\u0902 \u0938\u0930 \u092b\u094d\u0930\u093e\u0902\u0938\u093f\u0938 \u0917\u093e\u0932\u094d\u091f\u0928 (Sir Francis Galton) \u0928\u0947 \u0905\u092a\u0928\u0947 \u0936\u094b\u0927 \u0932\u0947\u0916 ‘\u092a\u0948\u0924\u0943\u0915 \u090a\u0901\u091a\u093e\u0908 \u092e\u0947\u0902 \u092e\u0927\u094d\u092f\u092e\u0924\u093e \u0915\u0940 \u0913\u0930 \u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928’ (Regression towards mediocrity in hereditary stature) \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u0925\u093e\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u00a0\u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 (Regression Analysis) \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940 \u0915\u093e \u090f\u0915 \u0905\u0924\u094d\u092f\u0902\u0924 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923\u093e\u0924\u094d\u092e\u0915 \u0909\u092a\u0915\u0930\u0923 \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0926\u094b \u092f\u093e \u0926\u094b \u0938\u0947 \u0905\u0927\u093f\u0915 \u091a\u0930 (Variables) \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 \u0928 \u0915\u0947\u0935\u0932 \u091a\u0930 \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u0940 \u0926\u093f\u0936\u093e \u090f\u0935\u0902 \u0924\u093e\u0915\u0924 \u0915\u094b \u092e\u093e\u092a\u0924\u093e \u0939\u0948, \u092c\u0932\u094d\u0915\u093f \u0907\u0938 \u0906\u0927\u093e\u0930 \u092a\u0930 \u092d\u0935\u093f\u0937\u094d\u092f \u0915\u0947 \u092e\u0942\u0932\u094d\u092f\u094b\u0902 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 (Prediction) \u092d\u0940 \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0940\u0932\u093f\u090f \u0907\u0938\u0947 \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u0915\u0947 \u0932\u093f\u090f \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u092d\u093e\u0935\u0940 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0935\u093f\u0927\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u092e\u0941\u0916\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0928\u093f\u0930\u094d\u092d\u0930 \u091a\u0930 (Dependent Variable) \u0914\u0930 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 (Independent Variable) \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u092f\u0926\u093f \u0939\u092e \u091c\u093e\u0928\u0928\u093e \u091a\u093e\u0939\u0947\u0902 \u0915\u093f \u0915\u093f\u0938\u0940 \u091b\u093e\u0924\u094d\u0930 \u0915\u0947 \u092a\u0930\u0940\u0915\u094d\u0937\u093e-\u0905\u0902\u0915 \u0909\u0938\u0915\u0940 \u092a\u0922\u093c\u093e\u0908 \u0915\u0947 \u0918\u0902\u091f\u094b\u0902 \u092a\u0930 \u0915\u0948\u0938\u0947 \u0928\u093f\u0930\u094d\u092d\u0930 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e Regression Analysis \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0917\u0947\u0964<\/span><\/p>\n

\u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u00a0 \u00a0<\/strong> <\/span>\u00a0<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u201cRegression \u0909\u0938 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0924\u0915\u0928\u0940\u0915 \u0915\u094b \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0938\u0915\u0947 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u094b \u091a\u0930 \u0915\u0947 \u092c\u0940\u091a \u092e\u093e\u0924\u094d\u0930\u093e\u0924\u094d\u092e\u0915 \u0938\u0902\u092c\u0902\u0927 \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u0909\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u091a\u0930 \u0915\u0947 \u092e\u0942\u0932\u094d\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964\u201d<\/span><\/p>\n

\u0938\u0915\u094d\u0937\u0947\u092a \u092e\u0947\u0902-<\/strong><\/span><\/p>\n

\u00a0 \u00a0 Regression \u0935\u0939 \u092a\u0926\u094d\u0927\u0924\u093f \u0939\u0948 \u091c\u093f\u0938\u092e\u0947\u0902 \u0939\u092e \u090f\u0915 \u091a\u0930 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0926\u0942\u0938\u0930\u0947 \u091a\u0930 \u0915\u0947 \u092e\u0942\u0932\u094d\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

Regression Analysis \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0914\u0930 \u0909\u092a\u092f\u094b\u0917\u093f\u0924\u093e<\/span><\/strong><\/p>\n

(i) \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 (Prediction):-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Regression Analysis \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0914\u0930 \u0909\u092a\u092f\u094b\u0917\u093f\u0924\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u092e\u0947\u0902 \u0907\u0938\u0932\u093f\u090f \u0939\u0948 \u0915\u093f \u092f\u0939 \u091a\u0930 (variables) \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930 \u092d\u0935\u093f\u0937\u094d\u092f \u0915\u0947 \u092e\u093e\u0928\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e\u0928\u0947, \u0930\u0941\u091d\u093e\u0928 \u092a\u0939\u091a\u093e\u0928\u0928\u0947 \u0914\u0930 \u0928\u093f\u0930\u094d\u0923\u092f-\u0928\u093f\u0930\u094d\u0927\u093e\u0930\u0923 \u092e\u0947\u0902 \u0938\u0939\u093e\u092f\u0924\u093e \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u092c\u093f\u0915\u094d\u0930\u0940, \u0909\u0924\u094d\u092a\u093e\u0926\u0928, \u092e\u093e\u0902\u0917, \u0924\u093e\u092a\u092e\u093e\u0928, GDP, \u091c\u0928\u0938\u0902\u0916\u094d\u092f\u093e \u0906\u0926\u093f \u0915\u093e \u092d\u0935\u093f\u0937\u094d\u092f \u0905\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e\u0928\u0947 \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917\u0940\u0964<\/span><\/p>\n

(ii) \u0928\u093f\u0930\u094d\u0923\u092f-\u0928\u093f\u0930\u094d\u092e\u093e\u0923 (Decision Making):-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 Regression Analysis \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0928\u093f\u0930\u094d\u0923\u092f-\u0928\u093f\u0930\u094d\u092e\u093e\u0923 \u092e\u0947\u0902 \u0907\u0938\u0932\u093f\u090f \u0939\u094b\u0924\u0940 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u092d\u0935\u093f\u0937\u094d\u092f \u0915\u0947 \u0930\u0941\u091d\u093e\u0928\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u092e\u093e\u0928, \u091a\u0930-\u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0936\u0940\u0932\u0924\u093e\u0913\u0902 \u0915\u093e \u092a\u094d\u0930\u092d\u093e\u0935 \u0914\u0930 \u0928\u0940\u0924\u093f\u0917\u0924 \u092f\u093e \u0935\u094d\u092f\u093e\u0935\u0938\u093e\u092f\u093f\u0915 \u0935\u093f\u0915\u0932\u094d\u092a\u094b\u0902 \u0915\u0940 \u0938\u091f\u0940\u0915\u0924\u093e \u0915\u094b \u092c\u0922\u093c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

(iv) \u091a\u0930-\u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928:-<\/span><\/strong><\/span><\/p>\n

\u00a0 Regression Analysis \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u091a\u0930-\u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u094b \u092e\u093e\u0924\u094d\u0930\u093e\u0924\u094d\u092e\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u092e\u091d\u0928\u0947, \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u092c\u0928\u093e\u0928\u0947, \u0915\u093e\u0930\u0923-\u092a\u094d\u0930\u092d\u093e\u0935 \u0926\u093f\u0936\u093e \u092a\u0939\u091a\u093e\u0928\u0928\u0947 \u0914\u0930 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930\u094b\u0902 \u0915\u093e \u0906\u0936\u094d\u0930\u093f\u0924 \u091a\u0930 \u092a\u0930 \u092a\u094d\u0930\u092d\u093e\u0935 \u092e\u093e\u092a\u0928\u0947 \u092e\u0947\u0902 \u0939\u094b\u0924\u0940 \u0939\u0948;<\/span><\/p>\n

(v) \u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u090f\u0935\u0902 \u0905\u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u092a\u094d\u0930\u092d\u093e\u0935\u094b\u0902 \u0915\u0940 \u092e\u093e\u092a<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0\u091c\u0948\u0938\u0947- \u0906\u092f \u0915\u093e \u0909\u092a\u092d\u094b\u0917 \u092a\u0930 \u092a\u094d\u0930\u092d\u093e\u0935, \u0936\u093f\u0915\u094d\u0937\u093e \u0915\u093e \u0906\u092f \u092a\u0930 \u092a\u094d\u0930\u092d\u093e\u0935\u0964<\/span><\/p>\n

(vi) \u0935\u0948\u091c\u094d\u091e\u093e\u0928\u093f\u0915 \u0935 \u0938\u093e\u092e\u093e\u091c\u093f\u0915 \u0905\u0928\u0941\u0938\u0902\u0927\u093e\u0928 \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u092e\u0928\u094b\u0935\u093f\u091c\u094d\u091e\u093e\u0928, \u091c\u0928\u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940, \u092d\u094c\u0924\u093f\u0915\u0940, \u091c\u0940\u0935\u0935\u093f\u091c\u094d\u091e\u093e\u0928, \u092d\u0942\u0917\u094b\u0932 \u0906\u0926\u093f \u092e\u0947\u0902 \u0935\u094d\u092f\u093e\u092a\u0915 \u0909\u092a\u092f\u094b\u0917\u0964<\/span><\/p>\n

Regression \u0915\u0947 \u092a\u094d\u0930\u0915\u093e\u0930 (Types of Regression Analysis)<\/strong><\/span><\/p>\n

1. Simple Regression (\u0938\u0930\u0932 \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928)<\/span><\/strong><\/span><\/h3>\n

\u00a0 \u00a0 \u00a0\u091c\u092c \u090f\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 \u0914\u0930 \u090f\u0915 \u0928\u093f\u0930\u094d\u092d\u0930 \u091a\u0930 \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u0909\u0926\u093e.\u2014\u0909\u092a\u092d\u094b\u0917 (Y) = f (\u0906\u092f X)<\/span><\/p>\n

Regression Equation: Y = a+bX<\/span><\/p>\n

2. Multiple Regression (\u092c\u0939\u0941-\u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928)<\/span><\/strong><\/span><\/h3>\n

\u091c\u092c \u0905\u0928\u0947\u0915 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 \u0915\u093f\u0938\u0940 \u090f\u0915 \u0928\u093f\u0930\u094d\u092d\u0930 \u091a\u0930 \u0915\u094b \u092a\u094d\u0930\u092d\u093e\u0935\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

\u0909\u0926\u093e.- \u0906\u092f (Y) = \u0936\u093f\u0915\u094d\u0937\u093e (X\u2081) + \u0905\u0928\u0941\u092d\u0935 (X\u2082) + \u0915\u094c\u0936\u0932 (X\u2083)<\/span><\/p>\n

Equation: Y=a+b1<\/sub>X1<\/sub>+b2<\/sub>X2<\/sub>+b3<\/sub>X3<\/sub>\u2026….<\/span><\/p>\n

3. Linear Regression (\u0930\u0947\u0916\u0940\u092f \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928)<\/span><\/strong><\/span><\/h3>\n

\u00a0 \u00a0 \u091c\u0939\u093e\u0901 \u0928\u093f\u0930\u094d\u092d\u0930 \u091a\u0930 \u0914\u0930 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0938\u0940\u0927\u0940 \u0930\u0947\u0916\u093e \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0939\u094b\u0964<\/span><\/p>\n

Y = a+bX<\/span><\/p>\n

Regression Line (\u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0930\u0947\u0916\u093e)<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 Regression Analysis \u0915\u093e \u092e\u0941\u0916\u094d\u092f \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u090f\u0915 \u0910\u0938\u0940 \u0930\u0947\u0916\u093e \u0916\u094b\u091c\u0928\u093e \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 \u0914\u0930 \u0928\u093f\u0930\u094d\u092d\u0930 \u091a\u0930 \u0915\u0947 \u092c\u0940\u091a \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0924\u094d\u0930\u0941\u091f\u093f (Minimum Error) \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u092c\u0902\u0927 \u0926\u0930\u094d\u0936\u093e\u090f\u0964<\/span><\/p>\n

Regression Coefficient (\u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0917\u0941\u0923\u093e\u0902\u0915)<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0Regression coefficient (b) \u092f\u0939 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u091a\u0930 \u092e\u0947\u0902 \u090f\u0915 \u0907\u0915\u093e\u0908 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0939\u094b\u0928\u0947 \u092a\u0930 \u0928\u093f\u0930\u094d\u092d\u0930 \u091a\u0930 \u092e\u0947\u0902 \u0915\u093f\u0924\u0928\u093e \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0939\u094b\u0917\u093e\u0964<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923:<\/span><\/strong><\/p>\n

\u092f\u0926\u093f b = 0.8<\/span><\/p>\n

\u0924\u094b X \u092e\u0947\u0902 1 \u0907\u0915\u093e\u0908 \u0935\u0943\u0926\u094d\u0927\u093f \u0939\u094b\u0928\u0947 \u092a\u0930 Y \u092e\u0947\u0902 0.8 \u0907\u0915\u093e\u0908 \u0935\u0943\u0926\u094d\u0927\u093f \u0939\u094b\u0917\u0940\u0964<\/span><\/p>\n

Regression coefficient \u0915\u0947 \u0917\u0941\u0923:<\/span><\/strong><\/p>\n

\u21d2 b \u0915\u093e \u092e\u093e\u0928 \u0927\u0928\u093e\u0924\u094d\u092e\u0915 (+) \u092f\u093e \u090b\u0923\u093e\u0924\u094d\u092e\u0915 (\u2013) \u0926\u094b\u0928\u094b\u0902 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u21d2 b = 0 \u092e\u0924\u0932\u092c \u0926\u094b\u0928\u094b\u0902 \u091a\u0930\u094b\u0902 \u092e\u0947\u0902 \u0915\u094b\u0908 \u0938\u0902\u092c\u0902\u0927 \u0928\u0939\u0940\u0902\u0964<\/span><\/p>\n

\u21d2 \u092f\u0926\u093f \u0926\u094b\u0928\u094b\u0902 b (bxy<\/sub> \u0914\u0930 byx<\/sub>) \u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0939\u094b\u0902 = \u0938\u0940\u0927\u093e \u0938\u0902\u092c\u0902\u0927\u0964<\/span><\/p>\n

\u21d2 \u092f\u0926\u093f \u0926\u094b\u0928\u094b\u0902 b \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u0939\u094b\u0902 = \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927\u0964<\/span><\/p>\n

Regression Analysis \u0915\u093e \u090f\u0915 \u0938\u0930\u0932 \u0909\u0926\u093e\u0939\u0930\u0923<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f\u0938\u0940 \u091b\u093e\u0924\u094d\u0930 \u0928\u0947 \u092a\u0922\u093c\u093e\u0908 (X) \u0915\u0947 \u0918\u0902\u091f\u0947 \u0914\u0930 \u092a\u0930\u0940\u0915\u094d\u0937\u093e-\u0905\u0902\u0915 (Y) \u0915\u093e \u0935\u093f\u0935\u0930\u0923 \u0926\u093f\u092f\u093e:<\/span><\/p>\n

\n
\n\n\n\n\n\n\n\n\n
X (\u0918\u0902\u091f\u0947)<\/span><\/td>\nY (\u0905\u0902\u0915)<\/span><\/td>\n<\/tr>\n<\/thead>\n
2<\/span><\/td>\n40<\/span><\/td>\n<\/tr>\n
3<\/span><\/td>\n50<\/span><\/td>\n<\/tr>\n
4<\/span><\/td>\n60<\/span><\/td>\n<\/tr>\n
5<\/span><\/td>\n70<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Regression Equation:<\/strong><\/span><\/p>\n

Y=a+bX<\/span><\/span><\/span><\/strong><\/span><\/p>\n

\u092f\u0926\u093f \u0917\u0923\u0928\u093e \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b:<\/strong><\/span><\/p>\n

a = 20, b = 10<\/strong><\/span><\/p>\n

\u0924\u094b \u0938\u092e\u0940\u0915\u0930\u0923 \u0939\u094b\u0917\u093e:<\/strong><\/span><\/p>\n

Y = 20+10X<\/span><\/span><\/span><\/strong><\/span><\/p>\n

\u092f\u0926\u093f X = 6 \u0918\u0902\u091f\u0947 \u092a\u0922\u093c\u093e\u0908 \u2192<\/strong><\/span><\/p>\n

Y = 20+10(6)=80<\/span><\/span><\/span><\/strong><\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924 \u091b\u093e\u0924\u094d\u0930 \u0905\u0928\u0941\u092e\u093e\u0928\u0924\u0903 80 \u0905\u0902\u0915 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0917\u093e\u0964<\/strong><\/span><\/p>\n

Correlation \u0914\u0930 Regression \u092e\u0947\u0902 \u0905\u0902\u0924\u0930<\/span><\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
\u092c\u093f\u0902\u0926\u0941<\/strong><\/span><\/td>\nCorrelation<\/strong><\/span><\/td>\nRegression<\/strong><\/span><\/td>\n<\/tr>\n
\u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f<\/span><\/td>\n\u0938\u0902\u092c\u0902\u0927 \u092e\u093e\u092a\u0928\u093e<\/span><\/td>\n\u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u0915\u0930\u0928\u093e<\/span><\/td>\n<\/tr>\n
\u0938\u0902\u092c\u0902\u0927 \u0926\u093f\u0936\u093e<\/span><\/td>\n\u0915\u0947\u0935\u0932 \u0926\u093f\u0936\u093e\/\u092e\u091c\u092c\u0942\u0924\u0940 \u092c\u0924\u093e\u0924\u093e \u0939\u0948<\/span><\/td>\n\u0906\u0936\u094d\u0930\u093f\u0924 \u0935 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0915\u093e \u0938\u0902\u092c\u0902\u0927 \u092c\u0924\u093e\u0924\u093e \u0939\u0948<\/span><\/td>\n<\/tr>\n
Variables<\/span><\/td>\n\u0915\u094b\u0908 dependent \u0928\u0939\u0940\u0902<\/span><\/td>\nY dependent, X independent<\/span><\/td>\n<\/tr>\n
\u092e\u093e\u0928 \u0938\u0940\u092e\u093e<\/span><\/td>\n-1 \u0938\u0947 +1<\/span><\/td>\n\u0915\u094b\u0908 \u092d\u0940 \u092e\u093e\u0928<\/span><\/td>\n<\/tr>\n
\u092a\u0930\u093f\u0923\u093e\u092e<\/span><\/td>\nr (\u0938\u0939\u0938\u0902\u092c\u0902\u0927 \u0917\u0941\u0923\u093e\u0902\u0915)<\/span><\/td>\nY = a + bX (\u092a\u094d\u0930\u0924\u093f\u0917\u092e\u0928 \u0938\u092e\u0940\u0915\u0930\u0923)<\/span><\/td>\n<\/tr>\n
\u0909\u092a\u092f\u094b\u0917<\/span><\/td>\n\u0938\u0902\u092c\u0902\u0927 \u0915\u0940 \u092e\u091c\u092c\u0942\u0924\u0940<\/span><\/td>\nPrediction (\u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
<\/div>\n
\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937<\/strong><\/span><\/div>\n

\u00a0 \u00a0 \u00a0\u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u090f\u0915 \u0905\u0924\u094d\u092f\u0902\u0924 \u092a\u094d\u0930\u092d\u093e\u0935\u0940 \u090f\u0935\u0902 \u0935\u0948\u091c\u094d\u091e\u093e\u0928\u093f\u0915 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0924\u0915\u0928\u0940\u0915 \u0939\u0948 \u091c\u094b \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u092e\u0947\u0902 \u092d\u0935\u093f\u0937\u094d\u092f \u0915\u0947 \u0905\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e\u0928\u0947, \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u0914\u0930 \u0928\u093f\u0930\u094d\u0923\u092f \u0932\u0947\u0928\u0947 \u092e\u0947\u0902 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092d\u0942\u092e\u093f\u0915\u093e \u0928\u093f\u092d\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0939 \u0915\u0947\u0935\u0932 \u0926\u094b \u091a\u0930\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927\u094b\u0902 \u0915\u093e \u092e\u093e\u0924\u094d\u0930\u093e\u0924\u094d\u092e\u0915 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0939\u0940 \u0928\u0939\u0940\u0902 \u0915\u0930\u0924\u093e, \u092c\u0932\u094d\u0915\u093f \u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0938\u092e\u093e\u0927\u093e\u0928 \u092d\u0940 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0\u0938\u091f\u0940\u0915 \u090f\u0935\u0902 \u0935\u093f\u0936\u094d\u0935\u0938\u0928\u0940\u092f \u0906\u0902\u0915\u0921\u093c\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0928\u0947 \u092a\u0930 Regression Analysis \u0936\u094b\u0927, \u0905\u0930\u094d\u0925\u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093e, \u0935\u093f\u091c\u094d\u091e\u093e\u0928, \u0938\u092e\u093e\u091c\u0936\u093e\u0938\u094d\u0924\u094d\u0930, \u0928\u0940\u0924\u093f-\u0928\u093f\u0930\u094d\u092e\u093e\u0923 \u0914\u0930 \u0935\u094d\u092f\u093e\u0935\u0938\u093e\u092f\u093f\u0915 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u092e\u0947\u0902 \u0905\u0924\u094d\u092f\u0902\u0924 \u0909\u092a\u092f\u094b\u0917\u0940 \u0938\u093f\u0926\u094d\u0927 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u0928\u094b\u091f: \u00a0\u0917\u093e\u0932\u094d\u091f\u0928 \u0928\u0947 \u0932\u0917\u092d\u0917 \u090f\u0915 \u0938\u0939\u0938\u094d\u0930 \u092a\u093f\u0924\u093e\u0913\u0902 \u0924\u0925\u093e \u092a\u0941\u0924\u094d\u0930\u094b \u0915\u0940 \u090a\u0901\u091a\u093e\u0907\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u0930\u0915\u0947 \u092f\u0939 \u091c\u094d\u091e\u093e\u0924 \u0915\u093f\u092f\u093e \u0915\u093f “\u0932\u092e\u094d\u092c\u0947 \u092a\u093f\u0924\u093e\u0913\u0902 \u0915\u0947 \u092a\u0941\u0924\u094d\u0930 \u092d\u0940 \u0932\u092e\u094d\u092c\u0947 \u0939\u094b\u0924\u0947 \u0939\u0948, \u0924\u0925\u093e \u091b\u094b\u091f\u0947 \u092a\u093f\u0924\u093e\u0913\u0902 \u0915\u0947 \u092a\u0941\u0924\u094d\u0930 \u092d\u0940 \u091b\u094b\u091f\u0947 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902, \u092a\u0930\u0928\u094d\u0924\u0941 \u0932\u092e\u094d\u092c\u0947 \u092a\u093f\u0924\u093e\u0913\u0902 \u0915\u0947 \u092a\u0941\u0924\u094d\u0930\u0940 \u0915\u0940 \u0914\u0938\u0924 \u090a\u0901\u091a\u093e\u0908 \u0909\u0928\u0915\u0947 \u092a\u093f\u0924\u093e\u0913\u0902 \u0915\u0940 \u090a\u0901\u091a\u093e\u0908 \u0915\u0940 \u0914\u0938\u0924 \u0938\u0947 \u0915\u092e \u0939\u094b\u0924\u0940 \u0939\u0948 \u0924\u0925\u093e \u091b\u094b\u091f\u0947 \u092a\u093f\u0924\u093e\u0913\u0902 \u0915\u0947 \u092a\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u090a\u0901\u091a\u093e\u0908 \u0909\u0928\u0915\u0947 \u092a\u093f\u0924\u093e\u0913\u0902 \u0915\u0940 \u090a\u0901\u091a\u093e\u0908 \u0915\u0940 \u0914\u0938\u0924 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964” <\/span><\/p>\n

\u00a0 \u00a0 \u00a0\u0917\u093e\u0932\u094d\u091f\u0928 \u0928\u0947 \u092e\u093e\u0928\u0935 \u091c\u093e\u0924\u093f \u092e\u0947\u0902 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u090a\u0901\u091a\u093e\u0908 (normal height) \u0915\u0940 \u0913\u0930 \u0932\u094c\u091f\u0928\u0947 \u0915\u0940 \u092a\u094d\u0930\u0935\u0943\u0924\u094d\u0924\u093f \u0915\u094b \u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928 \u0915\u094b \u0938\u0902\u091c\u094d\u091e\u093e \u0926\u0940 \u0939\u0948\u0964<\/span><\/p>\n

\n
\n
\n

\u0909\u0924\u094d\u0924\u0930 \u0932\u093f\u0916\u0928\u0947 \u0915\u093e \u0926\u0942\u0938\u0930\u093e \u0924\u0930\u0940\u0915\u093e<\/span><\/strong><\/span><\/p>\n

\n
\n
\n

Regression Analysis<\/span><\/strong><\/span><\/span><\/p>\n

Introduction:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0The mutual relationship between two Series is measured with the help of Correlation. Under correlation the direction and magnitude of the relationship between two variables is measured. But it is not possible to make the best estimate of the value of a dependent Variable on the basis of the given value of the Independent Variable by correlation analysis.<\/span><\/p>\n

\u00a0 \u00a0 \u00a0Therefor, to make the best estimates and future estimation, the Study of regression analysis is very Important useful.<\/span><\/p>\n

Meaning:<\/span><\/strong><\/p>\n

\u00a0 \u00a0 \u00a0Regression is the Study of the nature of relationship between that one may be able to Predict the Unknown value of one variable for a known value of another Variable.<\/span><\/p>\n

\u00a0 \u00a0 \u00a0In regression, one variable is considered as an independent Variable and another Variable is taken as dependent Variable. with the help of regression, possible values of the dependent Variable are estimated on the basis of values of the independent variable.<\/span><\/p>\n

\u00a0 \u00a0 \u00a0for example demand and price.<\/span><\/p>\n

On the basis of this relationship between demand and price probable Value of demand can be estimated Corresponding to the different Values of place.<\/span><\/p>\n

\u21d2 Regression Analysis is the technique for measuring or estimating the relationship among Variables.<\/span><\/p>\n

\u21d2 Regression Analysis provides estimates of Values of the dependent Variable From the Values of the independent Variable.<\/span><\/p>\n

\u21d2 The device used to accomplish this estimation procedure is the regression line.<\/span><\/p>\n

\u21d2 The regression line describes the average relationship existing between x and y.<\/span><\/p>\n

Exmple-<\/span><\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
x<\/strong><\/span><\/td>\ny<\/strong><\/span><\/td>\n<\/tr>\n
4<\/strong><\/span><\/td>\n100<\/strong><\/span><\/td>\n<\/tr>\n
7<\/strong><\/span><\/td>\n92<\/strong><\/span><\/td>\n<\/tr>\n
15<\/strong><\/span><\/td>\n90<\/strong><\/span><\/td>\n<\/tr>\n
18<\/strong><\/span><\/td>\n82<\/strong><\/span><\/td>\n<\/tr>\n
20<\/strong><\/span><\/td>\n79<\/strong><\/span><\/td>\n<\/tr>\n
25<\/strong><\/span><\/td>\n74<\/strong><\/span><\/td>\n<\/tr>\n
28<\/strong><\/span><\/td>\n68<\/strong><\/span><\/td>\n<\/tr>\n
.<\/strong><\/span><\/td>\n.<\/strong><\/span><\/td>\n<\/tr>\n
.<\/strong><\/span><\/td>\n.<\/strong><\/span><\/td>\n<\/tr>\n
35<\/strong><\/span><\/td>\n?\u00a0 \u00a0 (here, x given, y=?)<\/strong><\/span><\/td>\n<\/tr>\n
.<\/strong><\/span><\/td>\n.<\/strong><\/span><\/td>\n<\/tr>\n
.<\/strong><\/span><\/td>\n.<\/strong><\/span><\/td>\n<\/tr>\n
?\u00a0 (here, y given, x=?)<\/strong><\/span><\/td>\n20<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Difference Between Correlation and Regression<\/strong><\/span><\/p>\n\n\n\n\n\n\n\n\n\n
Correlation Analysis<\/strong><\/span><\/td>\nRegression Analysis<\/strong><\/span><\/td>\n<\/tr>\n
1. Correlation coefficient\u00a0 is a measure of degree of covariability between x and y.<\/span><\/td>\n1. Objective of regression analysis is to Study the nature of relationship between Variables.<\/span><\/td>\n<\/tr>\n
2. In Correlation analysis we can not say that one Variable is the cause & other the effect.<\/span><\/td>\n2. in regression analysis it is possible to study Cause & effect relationship.<\/span><\/td>\n<\/tr>\n
3. In Correlation analysis both rxy<\/sub> & ryx<\/sub> are Symmetric (rxy<\/sub>=ryx<\/sub>)<\/span><\/td>\n3. In regression analysis bxy<\/sub> & byx<\/sub> are not Symmetric. (bxy<\/sub>\u2260byx<\/sub>)<\/span><\/td>\n<\/tr>\n
4. There may be measure Correlation which has no practical relevance such as Increase in income & increase in weight.<\/span><\/td>\n4. There is nothing like non-Sense regression.<\/span><\/td>\n<\/tr>\n
5. Correlation coeff. is independent of change of scale and origin.\u00a0<\/span><\/td>\n5. Regression coefficientsare independent of change of origin but not of scale.<\/span><\/td>\n<\/tr>\n
\n

Similarity: Correlation coeff. (r) and regression coeff. (bxy<\/sub> & byx <\/sub>always have Same Sign.<\/span><\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

UTILITY OF REGRESSION<\/span><\/strong><\/span><\/span><\/p>\n

(i) Nature of Relationship:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 Regression analysis explains the nature of relationship between two variables.<\/span><\/p>\n

(ii) Estimation of Relationship:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0The mutual relationship between two or more variables can be measured easily by regression analysis.<\/span><\/p>\n

(iii) Prediction:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 By regression analysis, the value of a dependent variable can be predicted on the basis of the value of an independent variable. For example, if the price of a commodity rises, what will be the probable fall in demand, this can be predicted by regression.<\/span><\/p>\n

(iv) Useful in Economic and Business research:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 Regression analysis is very useful in business and economic research. With the help of regression, business and economic policies can be formulated.<\/span><\/p>\n

TYPES OF REGRESSION ANALYSIS<\/span><\/strong><\/h3>\n

1. Simple and Multiple Regression:-<\/span><\/strong><\/p>\n

\u00a0 \u00a0 \u00a0 In simple regression analysis, we study only two variables at the time, in which one variable is dependent and another is independent. For example income and expenditure.<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 on the contrary, we study more than two variables at a time in multiple, regression analysis in which one is dependent variable and others are independent variable. For example the study of rain and irrigation, the study of effect of rain and irrigation on yield of wheat.<\/span><\/p>\n

2. Linear and Non-Linear Regression:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 when one variable changes with other variable in some fixed ratio, this is called <\/span>linear regression. such type of relationship is depicted on a graph by means of straight line.<\/span><\/span><\/p>\n

\u00a0 \u00a0 On the other hand when one variable varies with other variable in a changing ratio, then it is referred to as curvi-linear or Non-linear regression. This relationship, expressed on a graph paper takes the form of curve.<\/span><\/p>\n

3. Partial and Total Regression:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0When two or more variables are studied for functional relationship but at a time relationship between only two variables is studied and other variables held constant then it is known as partial regression. On the other hand, in total regression all variables are studied simultaneous for the relationship among them.<\/span><\/p>\n

Simple Linear Regression<\/strong><\/span><\/p>\n

Regression lines: <\/strong><\/span><\/span><\/h3>\n

\u00a0 \u00a0 \u00a0Regression lines are linear equations that show the relationship between two variables. These help in making predictions, understanding trends and determining the direction of changes in the data.<\/span><\/p>\n\n\n\n\n\n\n\n\n\n
Regression lines<\/span><\/strong><\/td>\n<\/tr>\n
Regression line of y on x<\/span><\/strong><\/td>\nRegression line of x on y<\/span><\/strong><\/td>\n<\/tr>\n
1. y-\u0233 = byx <\/sub>(x-x\u0304)<\/span><\/p>\n

byx <\/sub>is regression coeff. of y on x<\/span><\/p>\n

byx <\/sub>=\"\"<\/span><\/td>\n

1. x-x\u0304 = bxy <\/sub>(y-\u0233)<\/span><\/p>\n

bxy<\/sub> is regression coeff. of x on y<\/span><\/p>\n

bxy <\/sub>= \"\"<\/span><\/td>\n<\/tr>\n

Both lines intersect at point (x\u0304, \u0233),\u00a0 x\u0304 = Mean of x, \u0233 = Mean of y<\/span><\/td>\n<\/tr>\n
2. x\u2192 Independent variable<\/span><\/p>\n

y\u2192 Dependent variable<\/span><\/td>\n

2. x\u2192Dependent variable<\/span><\/p>\n

y\u2192 Independent variable<\/span><\/td>\n<\/tr>\n

3. Used to calculate y for given x<\/span><\/td>\n3. Used to calculate y for given y<\/span><\/td>\n<\/tr>\n
4. Regression equation of y on x, y = a+bx<\/span><\/td>\n4. Regression equation of x on y, x = a+by<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Methods of Obtaining Regression Lines<\/span><\/strong><\/p>\n

(i) Scatter Diagram Method:-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 This is the simplest method of constructing regression lines, In this method, values of the related variables are plotted on a graph. A straight line is drawn passing through the plotted points. The straight line is drawn with free hand. This shape of regression line can be linear or non-linear also. This depends upon the location of plotted points. This method is very rarely used in practicle because in this method, the decision of the person who draws the regression lines very much affects the result.<\/span><\/p>\n

(ii) Least Square Method:-<\/span><\/strong><\/p>\n

\u00a0 \u00a0 Regression lines are also constructed by least Square method. Under this method, the two regression lines, is fitted through different points in such a way that the sum of squares of the deviations of the observed values from the fitted line shall be least. The line drawn by this method is called the Line of Best Fit. In other words, regression line of Y on X is so drawn such that vertically, the sum of squared deviation becomes minimum relating to the different points and the regression line on X on Y is so drawn such that horizontally, squared deviations of different points add up to minimum.<\/span><\/p>\n

Regression Lines <\/span><\/strong><\/span><\/p>\n

and <\/span><\/strong><\/span><\/p>\n

Degree of Correlation<\/span><\/strong><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

REGRESSION EQUATIONS<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0Regression equations are the algebraic formulation of regression lines. Regression equations present Regression lines.<\/span><\/p>\n

(i) Regression equation of x on y<\/span><\/strong><\/p>\n

\u00a0 \u00a0 \u00a0This equation is used to estimate the probable value of x on the basis of the given value of y.<\/span><\/p>\n

\u21d2 x-x\u0304 = bxy <\/sub>(y-\u0233)<\/span><\/strong><\/span><\/p>\n

(ii) Regression Equation of y on x<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0This equation is used to estimate the probable value of y on the basis of given value of x.<\/span><\/p>\n

\u21d2 y-\u0233 = byx <\/sub>(x-x\u0304)<\/span><\/strong><\/span><\/p>\n

REGRESSION COEFFICIENTS<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 Regression Coefficient measures the average change in the value of one variable for a unit change in the value of another variable. r Infact, it represents the slope of a regression line.<\/span><\/p>\n

(i) Regression Coefficient x on y<\/span><\/strong><\/p>\n

\u00a0 \u00a0 \u00a0This Coefficient shows that with the unit change in the value of y variable, what will be the average change in the value of x variable.<\/span><\/p>\n

bxy <\/sub>= r . \u03c3x\/\u03c3y<\/span><\/strong><\/p>\n

(ii) Regression coefficient of y on x<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 This Coefficient shows that with a unit change in the value of x variable, what will be the average change in the value of y variable.<\/span><\/p>\n

byx <\/sub>= r . \u03c3y\/\u03c3x<\/span><\/strong><\/p>\n

Example (i) Find both regression lines (x on y and y on x) for the following data.<\/span><\/strong><\/p>\n\n\n\n\n\n\n\n\n
x<\/span><\/strong><\/td>\ny<\/span><\/strong><\/td>\n<\/tr>\n
6<\/span><\/strong><\/span><\/td>\n9<\/span><\/strong><\/span><\/td>\n<\/tr>\n
2<\/span><\/strong><\/span><\/td>\n11<\/span><\/strong><\/span><\/td>\n<\/tr>\n
10<\/span><\/strong><\/span><\/td>\n5<\/span><\/strong><\/span><\/td>\n<\/tr>\n
4<\/span><\/strong><\/span><\/td>\n8<\/span><\/strong><\/span><\/td>\n<\/tr>\n
8<\/span><\/strong><\/span><\/td>\n7<\/span><\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculation:<\/span><\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
x<\/span><\/strong><\/td>\ny<\/span><\/strong><\/td>\nx2<\/sup><\/span><\/strong><\/td>\ny2<\/sup><\/span><\/strong><\/td>\nxy<\/span><\/strong><\/td>\n<\/tr>\n
6<\/span><\/strong><\/td>\n9<\/span><\/strong><\/td>\n36<\/span><\/strong><\/td>\n81<\/span><\/strong><\/td>\n54<\/span><\/strong><\/td>\n<\/tr>\n
2<\/span><\/strong><\/td>\n11<\/span><\/strong><\/td>\n4<\/span><\/strong><\/td>\n121<\/span><\/strong><\/td>\n22<\/span><\/strong><\/td>\n<\/tr>\n
10<\/span><\/strong><\/td>\n5<\/span><\/strong><\/td>\n100<\/span><\/strong><\/td>\n25<\/span><\/strong><\/td>\n50<\/span><\/strong><\/td>\n<\/tr>\n
4<\/span><\/strong><\/td>\n8<\/span><\/strong><\/td>\n16<\/span><\/strong><\/td>\n64<\/span><\/strong><\/td>\n32<\/span><\/strong><\/td>\n<\/tr>\n
8<\/span><\/strong><\/td>\n7<\/span><\/strong><\/td>\n64<\/span><\/strong><\/td>\n49<\/span><\/strong><\/td>\n56<\/span><\/strong><\/td>\n<\/tr>\n
30<\/span><\/strong><\/span><\/td>\n40<\/span><\/strong><\/span><\/td>\n220<\/span><\/strong><\/span><\/td>\n340<\/span><\/strong><\/span><\/td>\n214<\/span><\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

n = 5<\/span><\/strong><\/p>\n

x\u0304 = \u03a3x\/n = 30\/5 = 6<\/span><\/strong><\/p>\n

\u0233 = \u03a3y\/n = 40\/5 = 8<\/span><\/strong><\/p>\n

byx <\/sub>= \"\"<\/span><\/strong><\/p>\n

= 5×214 – 30×40 \/ 5×220 – 30×30<\/span><\/strong><\/p>\n

= -0.65<\/span><\/strong><\/p>\n

bxy<\/sub> = \"\"<\/span><\/strong><\/p>\n

= 5×214 – 30×40 \/ 5×340 – 40×40<\/span><\/strong><\/p>\n

= -1.3<\/span><\/strong><\/p>\n

now,<\/span><\/strong><\/p>\n

x on y \u21d2 x-x\u0304 = bxy <\/sub>(y-\u0233)<\/span><\/strong><\/p>\n

\u21d2 x-6 = -1.3(y-8)<\/span><\/strong><\/p>\n

\u21d2 x-6 = -1.3y + 1.3×8<\/span><\/strong><\/p>\n

\u21d2 x = -1.3y + 10.4 + 6<\/span><\/strong><\/p>\n

\u21d2 x = -1.3y + 16.4<\/span><\/strong><\/p>\n

y on x \u21d2 y-\u0233 = byx <\/sub>(x-x\u0304)<\/span><\/strong><\/p>\n

\u21d2 y-8 = -0.65 (x-6)<\/span><\/strong><\/p>\n

\u21d2 y-8 = -0.65x + 0.65×6<\/span><\/strong><\/p>\n

\u21d2 y = -0.65x + 3.9+8<\/span><\/strong><\/p>\n

\u21d2 y = -0.65x<\/a> + 11.9<\/a><\/span><\/strong><\/span><\/p>\n

Example (ii) From the following data:<\/span><\/strong><\/p>\n

(a) Obtain two regression lines<\/span><\/strong><\/p>\n

(b) Calculate correlation coeff.<\/span><\/strong><\/p>\n

(c) Estimate the value of y for x = 6.2<\/span><\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
x<\/span><\/strong><\/td>\ny<\/span><\/strong><\/td>\n<\/tr>\n
1<\/span><\/strong><\/td>\n9<\/span><\/strong><\/td>\n<\/tr>\n
2<\/span><\/strong><\/td>\n8<\/span><\/strong><\/td>\n<\/tr>\n
3<\/span><\/strong><\/td>\n10<\/span><\/strong><\/td>\n<\/tr>\n
4<\/span><\/strong><\/td>\n12<\/span><\/strong><\/td>\n<\/tr>\n
5<\/span><\/strong><\/td>\n11<\/span><\/strong><\/td>\n<\/tr>\n
6<\/span><\/strong><\/td>\n13<\/span><\/strong><\/td>\n<\/tr>\n
7<\/span><\/strong><\/td>\n14<\/span><\/strong><\/td>\n<\/tr>\n
8<\/span><\/strong><\/td>\n16<\/span><\/strong><\/td>\n<\/tr>\n
9<\/span><\/strong><\/td>\n15<\/span><\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculation:<\/strong><\/span><\/p>\n

(a) Obtain two regression lines<\/span><\/strong><\/p>\n

Note: x on y \u21d2 x-x\u0304 = bxy <\/sub>(y-\u0233)<\/span><\/p>\n

y on x \u21d2 y-\u0233 = byx <\/sub>(x-x\u0304)<\/span><\/p>\n

(i) For direct method\u00a0<\/span><\/strong><\/p>\n

We’ll need \u21d2 x\u0304, \u0233, bxy, byx<\/sub><\/span><\/p>\n

r = \"\"<\/span><\/p>\n

bxy <\/sub>= \"\"<\/span><\/p>\n

byx <\/sub>=\"\"<\/span><\/p>\n

Finally we’ll need \u21d2 n, \u03a3x, \u03a3y, \u03a3x2<\/sup>, \u03a3y2<\/sup>, \u03a3xy\u00a0<\/span><\/strong><\/p>\n

(ii) For Short-cut method :-<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0\u0938\u092e\u093e\u0928\u093e\u0928\u094d\u0924\u0930 \u092e\u093e\u0927\u094d\u092f \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092e\u0947\u0902 \u0939\u094b \u0924\u094b \u092a\u094d\u0930\u0924\u094d\u092f\u0915\u094d\u0937 \u0930\u0940\u0924\u093f \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928 \u0917\u0941\u0923\u093e\u0902\u0915 \u0938\u0930\u0932\u0924\u093e \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u092f\u0947 \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u0932\u0947\u0915\u093f\u0928 \u092f\u0926\u093f \u092f\u0947 \u092e\u093e\u0927\u094d\u092f \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0906\u092f\u0947 \u0924\u092c \u0932\u0918\u0941 \u0930\u0940\u0924\u093f \u0905\u092a\u0928\u093e\u0928\u093e \u0905\u0927\u093f\u0915 \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0939\u094b\u0917\u093e\u0964 \u0907\u0938 \u0930\u0940\u0924\u093f \u092e\u0947\u0902 \u0905\u0917\u094d\u0930 \u0938\u0942\u0924\u094d\u0930\u094b\u0902 \u0915\u0940 \u0938\u0939\u093e\u092f\u0924\u093e \u0938\u0947 \u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928 \u0917\u0941\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948:<\/span><\/p>\n

Note: x\u2192 dx = x-A,<\/sub><\/span><\/p>\n

y\u2192 <\/sub>dy = y-B<\/sub><\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
x<\/strong><\/span><\/td>\ny<\/strong><\/span><\/td>\ndx\u00a0<\/sub><\/strong><\/span><\/p>\n

(x-5)<\/span><\/td>\n

dy<\/sub><\/strong><\/span><\/p>\n

(y-12)<\/span><\/td>\n

dx2<\/sup>\u00a0<\/sub><\/strong><\/span><\/td>\ndy2<\/sup><\/sub><\/strong><\/span><\/td>\nd<\/strong>x <\/strong>d<\/strong>y<\/strong><\/sub><\/span><\/td>\n<\/tr>\n
1<\/strong><\/span><\/td>\n9<\/strong><\/span><\/td>\n-4<\/strong><\/span><\/td>\n-3<\/strong><\/span><\/td>\n16<\/strong><\/span><\/td>\n9<\/strong><\/span><\/td>\n12<\/strong><\/span><\/td>\n<\/tr>\n
2<\/strong><\/span><\/td>\n8<\/strong><\/span><\/td>\n-3<\/strong><\/span><\/td>\n-4<\/strong><\/span><\/td>\n9<\/strong><\/span><\/td>\n16<\/strong><\/span><\/td>\n12<\/strong><\/span><\/td>\n<\/tr>\n
3<\/strong><\/span><\/td>\n10<\/strong><\/span><\/td>\n-2<\/strong><\/span><\/td>\n-2<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n<\/tr>\n
4<\/strong><\/span><\/td>\n12<\/strong><\/span><\/td>\n-1<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n<\/tr>\n
5 (A)<\/strong><\/span><\/td>\n11 (B)<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n-1<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n<\/tr>\n
6<\/strong><\/span><\/td>\n13<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n1<\/strong><\/span><\/td>\n<\/tr>\n
7<\/strong><\/span><\/td>\n14<\/strong><\/span><\/td>\n2<\/strong><\/span><\/td>\n2<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n<\/tr>\n
8<\/strong><\/span><\/td>\n16<\/strong><\/span><\/td>\n3<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n9<\/strong><\/span><\/td>\n16<\/strong><\/span><\/td>\n12<\/strong><\/span><\/td>\n<\/tr>\n
9<\/strong><\/span><\/td>\n15<\/strong><\/span><\/td>\n4<\/strong><\/span><\/td>\n3<\/strong><\/span><\/td>\n16<\/strong><\/span><\/td>\n9<\/strong><\/span><\/td>\n12<\/strong><\/span><\/td>\n<\/tr>\n
45<\/strong><\/span><\/td>\n108<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n0<\/strong><\/span><\/td>\n60<\/strong><\/span><\/td>\n60<\/strong><\/span><\/td>\n57<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Regression Co-efficient of X on Y<\/strong><\/span><\/p>\n

\"\"<\/span><\/p>\n

bxy <\/sub>= 9(57) – 0(0) \/ 9(60) – (0)2<\/sup>\u00a0<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 = 0.95<\/span><\/p>\n

Regression Co-efficient of Y on X<\/strong><\/span><\/p>\n

\"\"<\/p>\n

byx <\/sub>= 9(57) – 0(0) \/ 9(60) – (0)2<\/sup><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 = 0.95<\/span><\/p>\n

x\u0304 = \u03a3x\/n = 45\/9 = 5<\/strong><\/span><\/p>\n

\u0233 = \u03a3y\/n = 108\/9 = 12<\/strong><\/span><\/p>\n

now,<\/strong><\/span><\/p>\n

x on y \u21d2 x-x\u0304 = bxy <\/sub>(y-\u0233)<\/strong><\/span><\/p>\n

\u21d2 x – 5 = 0.95(y-12)<\/span><\/p>\n

\u21d2 x – 5 = (0.95)y – 0.95(12)<\/span><\/p>\n

\u21d2 x = (0.95)y – 11.4 + 5<\/span><\/p>\n

\u21d2 x = (0.95)y – 6.4<\/span><\/p>\n

y on x \u21d2 y-\u0233 = byx <\/sub>(x-x\u0304)<\/strong><\/span><\/p>\n

\u21d2 y – 12 = 0.95(x – 5)<\/span><\/p>\n

\u21d2 y – 12 = (0.95)x – 0.95 (5)<\/span><\/p>\n

\u21d2 y – 12 = (0.95)x – 4.75<\/span><\/p>\n

\u21d2 y = (0.95)x – 4.75 + 12<\/span><\/p>\n

\u21d2 y = (0.95)x + 7.25<\/span><\/p>\n

(b) Calculate correlation coeff.<\/span><\/strong><\/span><\/p>\n

\"\"<\/span><\/p>\n

(c) Estimate the value of y for x = 6.2<\/span><\/strong><\/span><\/p>\n

y on x \u21d2 y-\u0233 = byx <\/sub>(x-x\u0304)<\/strong><\/span><\/p>\n

\u21d2 y = (0.95)x + 7.25<\/span><\/p>\n

\u21d2 y = (0.95)6.2 + 7.25<\/span><\/p>\n

\u21d2 y = 5.89 + 7.25<\/span><\/p>\n

\u21d2 y = 13.14<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Regression Analysis (\u092a\u094d\u0930\u0924\u0940\u0917\u092e\u0928 \u092f\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0917\u092e\u0928 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923) \u092a\u0930\u093f\u091a\u092f \u00a0 \u00a0 \u00a0 ‘\u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928’ (Regression) \u0936\u092c\u094d\u0926 \u0915\u093e \u0905\u0930\u094d\u0925 ‘\u092a\u0940\u091b\u0947 \u0932\u094c\u091f\u0928\u093e’ \u0905\u0925\u0935\u093e ‘\u0935\u093e\u092a\u0938 \u0906\u0928\u093e’ (to regress or return back) \u0939\u0948\u0964 \u0907\u0938 \u0936\u092c\u094d\u0926 \u0915\u093e \u0938\u0930\u094d\u0935\u092a\u094d\u0930\u0925\u092e \u092a\u094d\u0930\u092f\u094b\u0917 1877 \u092e\u0947\u0902 \u0938\u0930 \u092b\u094d\u0930\u093e\u0902\u0938\u093f\u0938 \u0917\u093e\u0932\u094d\u091f\u0928 (Sir Francis Galton) \u0928\u0947 \u0905\u092a\u0928\u0947 \u0936\u094b\u0927 \u0932\u0947\u0916 ‘\u092a\u0948\u0924\u0943\u0915 \u090a\u0901\u091a\u093e\u0908 \u092e\u0947\u0902 \u092e\u0927\u094d\u092f\u092e\u0924\u093e \u0915\u0940 \u0913\u0930 \u092a\u094d\u0930\u0924\u0940\u092a\u0917\u092e\u0928’ (Regression towards mediocrity in hereditary […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[70,72,162],"tags":[],"class_list":["post-20120","post","type-post","status-publish","format-standard","hentry","category-cartography","category-pg-semester-1","category-research-methodology"],"acf":{"test_duration":60,"q1_question":"","q1_option_a":"","q1_option_b":"","q1_option_c":"","q1_option_d":"","q1_correct":"","q1_explanation":"","q2_question":"","q2_option_a":"","q2_option_b":"","q2_option_c":"","q2_option_d":"","q2_correct":"","q2_explanation":"","q3_question":"","q3_option_a":"","q3_option_b":"","q3_option_c":"","q3_option_d":"","q3_correct":"","q3_explanation":"","q4_question":"","q4_option_a":"","q4_option_b":"","q4_option_c":"","q4_option_d":"","q4_correct":"","q4_explanation":"","q5_question":"","q5_option_a":"","q5_option_b":"","q5_option_c":"","q5_option_d":"","q5_correct":"","q5_explanation":"","q6_question":"","q6_option_a":"","q6_option_b":"","q6_option_c":"","q6_option_d":"","q6_correct":"","q6_explanation":"","q7_question":"","q7_option_a":"","q7_option_b":"","q7_option_c":"","q7_option_d":"","q7_correct":"","q7_explanation":"","q8_question":"","q8_option_a":"","q8_option_b":"","q8_option_c":"","q8_option_d":"","q8_correct":"","q8_explanation":"","q9_question":"","q9_option_a":"","q9_option_b":"","q9_option_c":"","q9_option_d":"","q9_correct":"","q9_explanation":"","q10_question":"","q10_option_a":"","q10_option_b":"","q10_option_c":"","q10_option_d":"","q10_correct":"","q10_explanation":"","q11_question":"","q11_option_a":"","q11_option_b":"","q11_option_c":"","q11_option_d":"","q11_correct":"","q11_explanation":"","q12_question":"","q12_option_a":"","q12_option_b":"","q12_option_c":"","q12_option_d":"","q12_correct":"","q12_explanation":"","q13_question":"","q13_option_a":"","q13_option_b":"","q13_option_c":"","q13_option_d":"","q13_correct":"","q13_explanation":"","q14_question":"","q14_option_a":"","q14_option_b":"","q14_option_c":"","q14_option_d":"","q14_correct":"","q14_explanation":"","q15_question":"","q15_option_a":"","q15_option_b":"","q15_option_c":"","q15_option_d":"","q15_correct":"","q15_explanation":"","q16_question":"","q16_option_a":"","q16_option_b":"","q16_option_c":"","q16_option_d":"","q16_correct":"","q16_explanation":"","q17_question":"","q17_option_a":"","q17_option_b":"","q17_option_c":"","q17_option_d":"","q17_correct":"","q17_explanation":"","q18_question":"","q18_option_a":"","q18_option_b":"","q18_option_c":"","q18_option_d":"","q18_correct":"","q18_explanation":"","q19_question":"","q19_option_a":"","q19_option_b":"","q19_option_c":"","q19_option_d":"","q19_correct":"","q19_explanation":"","q20_question":"","q20_option_a":"","q20_option_b":"","q20_option_c":"","q20_option_d":"","q20_correct":"","q20_explanation":"","q21_question":"","q21_option_a":"","q21_option_b":"","q21_option_c":"","q21_option_d":"","q21_correct":"","q21_explanation":"","q22_question":"","q22_option_a":"","q22_option_b":"","q22_option_c":"","q22_option_d":"","q22_correct":"","q22_explanation":"","q23_question":"","q23_option_a":"","q23_option_b":"","q23_option_c":"","q23_option_d":"","q23_correct":"","q23_explanation":"","q24_question":"","q24_option_a":"","q24_option_b":"","q24_option_c":"","q24_option_d":"","q24_correct":"","q24_explanation":"","q25_question":"","q25_option_a":"","q25_option_b":"","q25_option_c":"","q25_option_d":"","q25_correct":"","q25_explanation":"","q26_question":"","q26_option_a":"","q26_option_b":"","q26_option_c":"","q26_option_d":"","q26_correct":"","q26_explanation":"","q27_question":"","q27_option_a":"","q27_option_b":"","q27_option_c":"","q27_option_d":"","q27_correct":"","q27_explanation":"","q28_question":"","q28_option_a":"","q28_option_b":"","q28_option_c":"","q28_option_d":"","q28_correct":"","q28_explanation":"","q29_question":"","q29_option_a":"","q29_option_b":"","q29_option_c":"","q29_option_d":"","q29_correct":"","q29_explanation":"","q30_question":"","q30_option_a":"","q30_option_b":"","q30_option_c":"","q30_option_d":"","q30_correct":"","q30_explanation":"","q31_question":"","q31_option_a":"","q31_option_b":"","q31_option_c":"","q31_option_d":"","q31_correct":"","q31_explanation":"","q32_question":"","q32_option_a":"","q32_option_b":"","q32_option_c":"","q32_option_d":"","q32_correct":"","q32_explanation":"","q33_question":"","q33_option_a":"","q33_option_b":"","q33_option_c":"","q33_option_d":"","q33_correct":"","q33_explanation":"","q34_question":"","q34_option_a":"","q34_option_b":"","q34_option_c":"","q34_option_d":"","q34_correct":"","q34_explanation":"","q35_question":"","q35_option_a":"","q35_option_b":"","q35_option_c":"","q35_option_d":"","q35_correct":"","q35_explanation":"","q36_question":"","q36_option_a":"","q36_option_b":"","q36_option_c":"","q36_option_d":"","q36_correct":"","q36_explanation":"","q37_question":"","q37_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