One Sample T Test is used to test if the sample studied follows a normal distribution when its standard deviation is unknown and the mean is known. The mean is equal to the test value based on the sample drawn.

If the standard deviation is known for the assumed normal distribution, One Sample Z Test is used instead.

Requirements for using One Sample T Test:

The sample studied is assumed to follow a normal distribution.

The standard deviation of the assumed normal distribution is unknown.

The formula is:

where $X$ is the sample, ${\mathrm{\mu }}_0$ is the test value of mean, $s$ is the sample standard deviation, $N$ is the sample size, and $T$ follows Student's T distribution with $N-1$ degrees of freedom.

Determine the null hypothesis:

Null Hypothesis: ${\mathrm{\mu }}_0=15$ (the actual mean)

Substitute the information into the formula:

$t=\frac{\left(16-15\right)}{\left(\frac{5.144}{\sqrt{1000}}\right)}=6.1475$

Compute the p-value:

$p-\mathrm{value}=\mathrm{Probability}\left(\left|T\right|>6.1475\right)=\mathrm{Probability}\left(T<-6.1475\right)+\mathrm{Probability}\left(T>6.1475\right)=p=1.136158\cdot {10}^{-9}$, $T˜\mathrm{StudentT}\left(999\right)$

Draw the conclusion:

This statistical test provides evidence that the null hypothesis is false, so we reject the null hypothesis.