T-test for one sample Mean with critical values Calculator
Note: This calculator is for testing using sample values. If you only have sample mean and sd use ONE SAMPLE TEST FROM SAMPLE MEAN AND SD calculator
You can see a sample solution below. Enter your data to get the solution for your question
$$ \displaylines{---} $$
$$ \displaylines{First\;we\;have\;to\;find\;mean\;and\;sample\;standard\;deviation
} $$
$$ \displaylines{} $$
$$ \displaylines{Mean = \frac{\sum_{i=1}^{n}x_{i}}{n}
\\ \\ \,=\frac{2+4+6+7+8}{5}
\\ \\ \,=\frac{27}{5}
\\ \\Mean = 5.4
} $$
$$ x $$ | $$ \bar{x} $$ | $$ x-\bar{x} $$ | $$ [x-\bar{x}]^{2} $$ |
$$ 2 $$ | $$ 5.4 $$ | $$ -3.4 $$ | $$ 11.56 $$ |
$$ 4 $$ | $$ 5.4 $$ | $$ -1.4 $$ | $$ 1.96 $$ |
$$ 6 $$ | $$ 5.4 $$ | $$ 0.6 $$ | $$ 0.36 $$ |
$$ 7 $$ | $$ 5.4 $$ | $$ 1.6 $$ | $$ 2.56 $$ |
$$ 8 $$ | $$ 5.4 $$ | $$ 2.6 $$ | $$ 6.76 $$ |
$$ Total $$ | $$ $$ | $$ $$ | $$ 23.2 $$ |
$$ \displaylines{} $$
$$ \displaylines{Sample \;variance = (\sigma)^{2} = \frac{\sum_{i=0}^{n}(x_{i}-\bar{x})^{2}}{n-1}
\\ \\ \Rightarrow
\frac{23.2}{4}
\\ \\ \Rightarrow
5.8
\\ \\ \sigma = \sqrt{variance}
\\ \\ \Rightarrow
\mathbf{\color{Red}{2.408319}}
} $$
$$ \displaylines{} $$
$$ \displaylines{\\ \\
Now\;we\;can\;start\;significant\;test
\\ \\
} $$
$$ \displaylines{} $$
$$ \displaylines{ \mathbf{\color{Red}{Right\;tail\;test}}
\\ \\
\\ \\
\bar{x} = 5.4
\\ \\
\mu = 5
\\ \\
s = 2.408319
\\ \\
n = 5
\\ \\
\alpha = 0.05
\\ \\
H_{0}: \mu = 5
\\ \\
H_{a}: \mu > 5
\\ \\
df=n-1=
5 -1
\\ \\ \Rightarrow
4
\\ \\
critical\;value\;t\;=\;\;\mathbf{\color{Red}{2.131847}}
\\\;\\\;Formula\;for\;test\;statistic\;t\;is
\\ \\ t=
\frac{\bar{x} - \mu }{\frac{s }{\sqrt{n}}}
\\ \\ \\ So, t = \frac{5.4-5}{\frac{2.408319}{\sqrt{5}}}
\\ \\ \Rightarrow
\frac{0.4}{1.077033}
\\ \\ \Rightarrow
0.371391
\\ \\
test\;statistic\;= \mathbf{\color{Red}{0.371391}}
\\ \\
p= 0.364591
\\ \\
p\;is\;more\;than\;\alpha\;.\;
So,\;Failed\;to\;Reject\;H_{0}
} $$
$$ \displaylines{} $$
$$ \displaylines{} $$