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# T-test for two sample assuming equal variances Calculator

Note: This calculator is for testing using sample values. If you only have sample mean and sd TWO SAMPLE TEST FROM SAMPLE MEAN AND SD calculator

Enter sample 1 values Use ',' or new line to separate between values: 2, 4, 6, 7, 8 3, 5, 6, 7, 5, 9 One tailed Two tailed

## You can see a sample solution below. Enter your data to get the solution for your question

$$\displaylines{---}$$
$$\displaylines{First\;we\;have\;to\;find\;mean\;and\;sample\;standard\;deviation\;of\;sample\;1 \\ \\ }$$
$$\displaylines{}$$
$$\displaylines{Mean = \frac{\sum_{i=1}^{n}X1_{i}}{n} \\ \\ \,=\frac{2+4+6+7+8}{5} \\ \\ \,=\frac{27}{5} \\ \\Mean = 5.4 }$$
$$x$$
$$\bar{x}$$
$$x-\bar{x}$$
$$[x-\bar{x}]^{2}$$
$$2$$
$$5.4$$
$$-3.4$$
$$11.56$$
$$4$$
$$5.4$$
$$-1.4$$
$$1.96$$
$$6$$
$$5.4$$
$$0.6$$
$$0.36$$
$$7$$
$$5.4$$
$$1.6$$
$$2.56$$
$$8$$
$$5.4$$
$$2.6$$
$$6.76$$
$$Total$$


$$23.2$$
$$\displaylines{}$$
$$\displaylines{Sample \;variance = (\sigma)^{2} = \frac{\sum_{i=0}^{n}(x_{i}-\bar{x})^{2}}{n-1} \\ \\ \Rightarrow \frac{23.2}{4} \\ \\ \Rightarrow 5.8 \\ \\ \sigma = \sqrt{variance} \\ \\ \Rightarrow \mathbf{\color{Red}{2.408319}} }$$
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$$\displaylines{Now,\;we\;have\;to\;find\;mean\;and\;sample\;standard\;deviation\;of\;sample\;2 \\ \\ }$$
$$\displaylines{}$$
$$\displaylines{Mean = \frac{\sum_{i=1}^{n}X2_{i}}{n} \\ \\ \,=\frac{3+5+6+7+5+9}{6} \\ \\ \,=\frac{35}{6} \\ \\Mean = 5.833333 }$$
$$x$$
$$\bar{x}$$
$$x-\bar{x}$$
$$[x-\bar{x}]^{2}$$
$$3$$
$$5.833333$$
$$-2.833333$$
$$8.027776$$
$$5$$
$$5.833333$$
$$-0.833333$$
$$0.694444$$
$$6$$
$$5.833333$$
$$0.166667$$
$$0.027778$$
$$7$$
$$5.833333$$
$$1.166667$$
$$1.361112$$
$$5$$
$$5.833333$$
$$-0.833333$$
$$0.694444$$
$$9$$
$$5.833333$$
$$3.166667$$
$$10.02778$$
$$Total$$


$$20.833334$$
$$\displaylines{}$$
$$\displaylines{Sample \;variance = (\sigma)^{2} = \frac{\sum_{i=0}^{n}(x_{i}-\bar{x})^{2}}{n-1} \\ \\ \Rightarrow \frac{20.833334}{5} \\ \\ \Rightarrow 4.166667 \\ \\ \sigma = \sqrt{variance} \\ \\ \Rightarrow \mathbf{\color{Red}{2.041241}} }$$
$$\displaylines{}$$
$$\displaylines{Now\;we\;can\;start\;significant\;test \\ \\ }$$
$$\displaylines{}$$
$$\displaylines{ \mathbf{\color{Red}{left\;tail\;test}} \\ \\ x1 = 5.4 \\ \\ x2 = 5.833333 \\ \\ n1 = 5 \\ \\ n2 = 6 \\ \\ s1 = 2.408319 \\ \\ s2 = 2.041241 \\ \\ \alpha = 0.05 \\ \\ H_{0}: \mu_{1} < \mu_{2} \\ \\ H_{a}: H_{0} is false \\ \\ Assuming\;equal\;variances,\;So\;pooled\;standard\;deviation\;=\;s_{p} \\ \\ \Rightarrow \sqrt{\frac{(n_{1}-1)s_{1}^{2} + (n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}} \\ \\ \Rightarrow \sqrt{\frac{(5-1)2.408319^{2} + (6-1)2.041241^{2}}{5+6-2}} \\ \\ \Rightarrow \sqrt{\frac{44.033326}{9}} \\ \\ \Rightarrow \sqrt{4.892592} \\ \\ \Rightarrow 2.21192 \\ \\ t= \frac{x1-x2}{s_{p} \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}} \\ \\ \\ \\ \Rightarrow \frac{5.4-5.833333}{2.21192 \sqrt{\frac{1}{5}+\frac{1}{6}}} \\ \\ \\ \\ \Rightarrow \frac{-0.433333}{1.339384} \\ \\ \\ \\ \Rightarrow -0.323532 \\ \\ df=n1+n2-2= 5 + 6 -2 \\ \\ \Rightarrow 9 \\ \\ Critical\;values\;are\; 2.262157 -2.262157 \\ \\ p= 0.376842 \\ \\ The\;result\;is\;not\;significant\;at\;p\;<\;0.05.\;So\;Failed\;to\;Reject\;H_{0} }$$
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