{"id":21935,"date":"2026-09-05T22:35:42","date_gmt":"2026-09-05T17:05:42","guid":{"rendered":"https:\/\/www.geographynotespdf.com\/?p=21935"},"modified":"2026-09-10T20:01:19","modified_gmt":"2026-09-10T14:31:19","slug":"students-t-test","status":"publish","type":"post","link":"https:\/\/geographynotespdf.com\/cms\/students-t-test\/","title":{"rendered":"Students\u2019 t-Test (\u0938\u094d\u091f\u0942\u0921\u0947\u0902\u091f\u094d\u0938 t-\u092a\u0930\u0940\u0915\u094d\u0937\u0923) | Formula, Types, Hypothesis, Degrees of Freedom & Applications | Statistics & Research Methodology"},"content":{"rendered":"\n\n\n
\n

Student’s t-Test<\/strong> <\/span><\/h2>\n

\u0938\u094d\u091f\u0942\u0921\u0947\u0902\u091f\u094d\u0938 t-\u092a\u0930\u0940\u0915\u094d\u0937\u0923<\/span><\/strong><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

\u00a0 \u00a0 \u00a0Students\u2019 t-Test<\/strong> \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 parametric statistical test<\/strong> \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u091b\u094b\u091f\u0947 sample (small sample) \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0926\u094b \u092e\u093e\u0927\u094d\u092f\u094b\u0902 (means) \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0915\u0940 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u092e\u0939\u0924\u094d\u0924\u093e (statistical significance)<\/strong> \u091c\u093e\u0901\u091a\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 <\/span><\/p>\n

\u00a0 \u00a0\u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u091c\u092c population \u0915\u093e standard deviation \u091c\u094d\u091e\u093e\u0924 \u0928 \u0939\u094b \u0914\u0930 sample size \u091b\u094b\u091f\u093e \u0939\u094b, \u0924\u092c t-test \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0\u092d\u0942\u0917\u094b\u0932, \u0938\u092e\u093e\u091c\u0936\u093e\u0938\u094d\u0924\u094d\u0930, \u0905\u0930\u094d\u0925\u0936\u093e\u0938\u094d\u0924\u094d\u0930 \u0924\u0925\u093e \u0905\u0928\u094d\u092f \u0938\u093e\u092e\u093e\u091c\u093f\u0915 \u0935\u093f\u091c\u094d\u091e\u093e\u0928\u094b\u0902 \u092e\u0947\u0902 \u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0926\u094b \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902, \u0926\u094b \u0938\u092e\u0942\u0939\u094b\u0902 \u092f\u093e \u0915\u093f\u0938\u0940 \u0918\u091f\u0928\u093e \u0915\u0947 \u0926\u094b \u0938\u092e\u092f\u094b\u0902 \u0915\u0947 \u0914\u0938\u0924 \u092e\u093e\u0928\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0 Students\u2019 t-Test \u0915\u0940 \u092e\u0941\u0916\u094d\u092f \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u090f\u0901<\/span><\/p>\n

✍️ <\/span><\/span>\u092f\u0939 \u090f\u0915 Parametric Test<\/strong> \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u0907\u0938\u0915\u093e \u0906\u0927\u093e\u0930 t-distribution<\/strong> \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u091b\u094b\u091f\u0947 sample \u0915\u0947 \u0932\u093f\u090f \u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u0907\u0938\u092e\u0947\u0902 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 population variance \u0905\u091c\u094d\u091e\u093e\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u092e\u0941\u0916\u094d\u092f\u0924\u0903 Mean \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/strong> \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u092f\u0939 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948 \u0915\u093f observed difference \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0939\u0948 \u092f\u093e \u0915\u0947\u0935\u0932 sampling variation \u0915\u0947 \u0915\u093e\u0930\u0923 \u0939\u0948\u0964<\/span><\/p>\n

t-Test \u0915\u0947 \u092a\u094d\u0930\u092e\u0941\u0916 \u092a\u094d\u0930\u0915\u093e\u0930<\/strong><\/span><\/p>\n

\u00a0 \u00a0 Students\u2019 t-Test \u0915\u094b \u092e\u0941\u0916\u094d\u092f\u0924\u0903 \u0924\u0940\u0928 \u092a\u094d\u0930\u0915\u093e\u0930\u094b\u0902 \u092e\u0947\u0902 \u0938\u092e\u091d\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/span><\/p>\n

1. One-Sample t-Test:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0\u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u093f\u0938\u0940 \u090f\u0915 sample \u0915\u0947 mean \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u093f\u0938\u0940 \u091c\u094d\u091e\u093e\u0924 \u092f\u093e hypothesized population mean<\/strong> \u0938\u0947 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923: <\/strong><\/span>\u0915\u093f\u0938\u0940 \u091c\u093f\u0932\u0947 \u092e\u0947\u0902 \u0915\u093f\u0938\u093e\u0928\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u0935\u093e\u0930\u094d\u0937\u093f\u0915 \u0906\u092f \u20b980,000 \u0939\u0948 \u092f\u093e \u0928\u0939\u0940\u0902, \u0907\u0938\u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0947 \u0932\u093f\u090f sample farmers \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0928\u093e\u0964<\/span><\/span><\/p>\n

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

t = (X\u0304 \u2212 \u03bc) \/ (s \/ \u221an)<\/strong><\/span><\/p>\n

\\[ t = \\frac{\\bar{X} – \\mu}{s\/\\sqrt{n}} \\]<\/span><\/p>\n

\u091c\u0939\u093e\u0901-<\/span><\/p>\n

X\u0304<\/strong><\/span> = Sample Mean<\/span><\/p>\n

\u03bc<\/strong>\u00a0= Population\/Hypothesized Mean<\/span><\/p>\n

s = Sample Standard Deviation<\/span><\/p>\n

n<\/span> = Sample Size<\/span><\/p>\n

Degrees of Freedom:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 df = n-1<\/strong><\/span><\/p>\n

2. Independent-Samples t-Test:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0926\u094b \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0938\u092e\u0942\u0939\u094b\u0902 (independent groups)<\/strong> \u0915\u0947 means \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923: <\/strong><\/span>\u0936\u0939\u0930\u0940 \u0914\u0930 \u0917\u094d\u0930\u093e\u092e\u0940\u0923 \u092a\u0930\u093f\u0935\u093e\u0930\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u0906\u092f \u092e\u0947\u0902 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0905\u0902\u0924\u0930 \u0939\u0948 \u092f\u093e \u0928\u0939\u0940\u0902\u0964<\/span><\/span><\/p>\n

\u00a0 \u00a0\u0907\u0938\u092e\u0947\u0902 \u0926\u094b\u0928\u094b\u0902 samples \u0905\u0932\u0917-\u0905\u0932\u0917 \u0914\u0930 independent \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923: <\/strong><\/span>\u0917\u094d\u0930\u093e\u092e\u0940\u0923 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0947 30 \u092a\u0930\u093f\u0935\u093e\u0930 \u0914\u0930 <\/span>\u0936\u0939\u0930\u0940 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0947 30 \u092a\u0930\u093f\u0935\u093e\u0930<\/span><\/span><\/p>\n

\u00a0 \u00a0\u0926\u094b\u0928\u094b\u0902 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u0906\u092f \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f independent t-test \u0932\u0917\u093e\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

3. Paired-Samples t-Test:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0924\u092c \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u092c observations paired \u092f\u093e related<\/strong> \u0939\u094b\u0902\u0964 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 \u090f\u0915 \u0939\u0940 sample \u0915\u0947 Before \u0914\u0930 After<\/strong> values \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923: <\/strong><\/span>\u0915\u093f\u0938\u0940 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u092e\u0947\u0902 \u0938\u093f\u0902\u091a\u093e\u0908 \u0938\u0941\u0935\u093f\u0927\u093e \u0932\u093e\u0917\u0942 \u0939\u094b\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947 \u0914\u0930 \u092c\u093e\u0926 \u092e\u0947\u0902 \u0915\u093f\u0938\u093e\u0928\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u0915\u0943\u0937\u093f \u0909\u0924\u094d\u092a\u093e\u0926\u0915\u0924\u093e \u0915\u0940 \u0924\u0941\u0932\u0928\u093e\u0964<\/span><\/span><\/p>\n

\u092f\u0939\u093e\u0901:<\/span><\/p>\n

Before \u2192 Intervention \u2192 After<\/strong><\/span><\/p>\n

\u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964<\/span><\/p>\n

t-Test \u092e\u0947\u0902 Hypothesis:<\/strong><\/span><\/p>\n

Null Hypothesis (H\u2080)<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u0926\u094b\u0928\u094b\u0902 means \u0915\u0947 \u092c\u0940\u091a \u0915\u094b\u0908 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0905\u0902\u0924\u0930 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/span><\/p>\n

H0 : \u03bc1 = \u03bc2<\/span><\/p>\n

Alternative Hypothesis (H\u2081)<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u0926\u094b\u0928\u094b\u0902 means \u0915\u0947 \u092c\u0940\u091a \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0905\u0902\u0924\u0930 \u0939\u0948\u0964<\/strong><\/span><\/p>\n

H1 : \u03bc1 \u2260 \u03bc2<\/span><\/p>\n

\u00a0 \u00a0 \u00a0\u092f\u0926\u093f \u0936\u094b\u0927\u0915\u0930\u094d\u0924\u093e directional hypothesis \u0930\u0916\u0924\u093e \u0939\u0948, \u0924\u094b:<\/span><\/p>\n

H1 : \u03bc1>\u03bc2<\/span><\/p>\n

\u092f\u093e<\/span><\/p>\n

H1 : \u03bc1<\u03bc2<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

Degrees of Freedom<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0Degrees of Freedom (df)<\/strong> \u092f\u0939 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f statistical test \u092e\u0947\u0902 \u0915\u093f\u0924\u0928\u0947 independent observations \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0930\u0942\u092a \u0938\u0947 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

One-sample t-test \u092e\u0947\u0902:<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 df =n\u22121<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923:<\/span>\u00a0\u092f\u0926\u093f sample size n = 20<\/span>\u00a0\u0939\u0948, \u0924\u094b:<\/span><\/p>\n

\u00a0 \u00a0 \u00a0df = 20\u22121 = 19<\/span><\/p>\n

t-Test \u0915\u0940 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 Research Problem \u2192 Hypothesis \u2192 Data Collection \u2192 Mean Calculation \u2192 t-value Calculation \u2192 Critical Value\/p-value \u2192 Decision<\/span><\/strong><\/span><\/p>\n

Step 1: Hypothesis \u092c\u0928\u093e\u0928\u093e<\/span><\/p>\n

H\u2080 \u0914\u0930 H\u2081 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n

Step 2: Significance Level \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u093e<\/span><\/p>\n

\u00a0 \u00a0 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903: <\/span>\u03b1 = 0.05 <\/span>\u0932\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/span><\/p>\n

Step 3: t-value \u0928\u093f\u0915\u093e\u0932\u0928\u093e<\/span><\/p>\n

\u0909\u092a\u092f\u0941\u0915\u094d\u0924 t-test formula \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n

Step 4: Degrees of Freedom \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u093e<\/span><\/p>\n

Step 5: Decision \u0932\u0947\u0928\u093e<\/span><\/p>\n

\u092f\u0926\u093f p-value < 0.05<\/strong>, \u0924\u094b \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 H\u2080 \u0915\u094b reject \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u092f\u0926\u093f p-value > 0.05<\/strong>, \u0924\u094b H\u2080 \u0915\u094b reject \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 evidence \u0928\u0939\u0940\u0902 \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e\u0964<\/span><\/p>\n

t-Test \u0915\u0940 \u092a\u094d\u0930\u092e\u0941\u0916 Assumptions:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 t-test \u0932\u0917\u093e\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947 \u0915\u0941\u091b assumptions \u0915\u093e \u0927\u094d\u092f\u093e\u0928 \u0930\u0916\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948:<\/span><\/p>\n

1. Data quantitative \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n

2. Observations \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 independent \u0939\u094b\u0928\u0947 \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n

3. Data approximately normally distributed \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f, \u0935\u093f\u0936\u0947\u0937\u0915\u0930 \u091b\u094b\u091f\u0947 samples \u092e\u0947\u0902\u0964<\/span><\/p>\n

4. Independent t-test \u092e\u0947\u0902 variance \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964<\/span><\/p>\n

5. Sample random \u092f\u093e \u0909\u091a\u093f\u0924 sampling procedure \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n

6. Extreme outliers \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u094b \u092a\u094d\u0930\u092d\u093e\u0935\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

\u092d\u0942\u0917\u094b\u0932 \u092e\u0947\u0902 t-Test \u0915\u0947 Applications<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0\u092d\u094c\u0917\u094b\u0932\u093f\u0915 \u0936\u094b\u0927 \u092e\u0947\u0902 t-test \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948-<\/span><\/p>\n

✍️ <\/span><\/span>\u0926\u094b \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u0935\u0930\u094d\u0937\u093e \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0926\u094b \u091c\u093f\u0932\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u0915\u0943\u0937\u093f \u0909\u0924\u094d\u092a\u093e\u0926\u0915\u0924\u093e \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0917\u094d\u0930\u093e\u092e\u0940\u0923 \u090f\u0935\u0902 \u0936\u0939\u0930\u0940 \u0906\u092f \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0926\u094b \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u0915\u0940 \u0914\u0938\u0924 \u091c\u0928\u0938\u0902\u0916\u094d\u092f\u093e \u0935\u0943\u0926\u094d\u0927\u093f \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0938\u093f\u0902\u091a\u093e\u0908 \u0938\u0947 \u092a\u0939\u0932\u0947 \u0914\u0930 \u092c\u093e\u0926 \u0915\u0940 \u0915\u0943\u0937\u093f \u0909\u0924\u094d\u092a\u093e\u0926\u0915\u0924\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0926\u094b \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0914\u0938\u0924 \u0924\u093e\u092a\u092e\u093e\u0928 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0926\u094b \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u0947 \u092d\u0942\u092e\u093f-\u0909\u092a\u092f\u094b\u0917 \u092a\u0948\u091f\u0930\u094d\u0928 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e<\/span><\/p>\n

✍️ <\/span><\/span>\u0935\u093f\u0915\u093e\u0938 \u0915\u093e\u0930\u094d\u092f\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u0939\u0932\u0947 \u0914\u0930 \u092c\u093e\u0926 \u0915\u0947 \u0938\u093e\u092e\u093e\u091c\u093f\u0915-\u0906\u0930\u094d\u0925\u093f\u0915 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928<\/span><\/p>\n

t-Test \u0915\u0947 Advantages:<\/strong><\/span><\/p>\n

✍️ <\/span><\/span>\u091b\u094b\u091f\u0947 sample \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917\u0940\u0964<\/span><\/p>\n

✍️ <\/span><\/span>Mean comparison \u0915\u0947 \u0932\u093f\u090f \u092a\u094d\u0930\u092d\u093e\u0935\u0940\u0964<\/span><\/p>\n

✍️ <\/span><\/span>Statistical significance \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0938\u0939\u093e\u092f\u0924\u093e \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>Social Science \u0914\u0930 Geography research \u092e\u0947\u0902 \u0935\u094d\u092f\u093e\u092a\u0915 \u0909\u092a\u092f\u094b\u0917\u0964<\/span><\/p>\n

✍️ <\/span><\/span>Calculation \u0905\u092a\u0947\u0915\u094d\u0937\u093e\u0915\u0943\u0924 \u0938\u0930\u0932 \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>One-sample, independent \u0914\u0930 paired data \u0915\u0947 \u0932\u093f\u090f \u0905\u0932\u0917-\u0905\u0932\u0917 \u0930\u0942\u092a \u0909\u092a\u0932\u092c\u094d\u0927 \u0939\u0948\u0902\u0964<\/span><\/p>\n

t-Test \u0915\u0940 Limitations:<\/span><\/strong><\/span><\/p>\n

✍️ <\/span><\/span>\u0907\u0938\u0915\u0940 assumptions \u092a\u0942\u0930\u0940 \u0939\u094b\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u0905\u0924\u094d\u092f\u0927\u093f\u0915 skewed data \u092e\u0947\u0902 \u092a\u0930\u093f\u0923\u093e\u092e \u092a\u094d\u0930\u092d\u093e\u0935\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>Outliers t-value \u0915\u094b \u092a\u094d\u0930\u092d\u093e\u0935\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u092f\u0939 \u092e\u0941\u0916\u094d\u092f\u0924\u0903 quantitative data \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>\u0917\u0932\u0924 test selection \u0938\u0947 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0917\u0932\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n

\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 Student’s t-Test<\/strong> \u091b\u094b\u091f\u0947 sample \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 quantitative data \u092e\u0947\u0902 means \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0915\u0940 statistical significance \u091c\u093e\u0901\u091a\u0928\u0947 \u0915\u0940 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0935\u093f\u0927\u093f \u0939\u0948\u0964 <\/span><\/p>\n

\u00a0 \u00a0 \u00a0One-sample, independent \u0924\u0925\u093e paired t-test \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u092d\u094c\u0917\u094b\u0932\u093f\u0915 \u090f\u0935\u0902 \u0938\u093e\u092e\u093e\u091c\u093f\u0915-\u0906\u0930\u094d\u0925\u093f\u0915 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964 \u0936\u094b\u0927 \u092e\u0947\u0902 \u0909\u091a\u093f\u0924 assumptions, hypothesis, degrees of freedom \u0924\u0925\u093e significance level \u0915\u093e \u092a\u093e\u0932\u0928 \u0915\u0930\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948\u0964<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Student’s t-Test \u0938\u094d\u091f\u0942\u0921\u0947\u0902\u091f\u094d\u0938 t-\u092a\u0930\u0940\u0915\u094d\u0937\u0923 \u092a\u0930\u093f\u091a\u092f: \u00a0 \u00a0 \u00a0Students\u2019 t-Test \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 parametric statistical test \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u091b\u094b\u091f\u0947 sample (small sample) \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0926\u094b \u092e\u093e\u0927\u094d\u092f\u094b\u0902 (means) \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0915\u0940 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u092e\u0939\u0924\u094d\u0924\u093e (statistical significance) \u091c\u093e\u0901\u091a\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u00a0 \u00a0\u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947 \u091c\u092c population \u0915\u093e standard 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