13.1 One-Sample: Hypothesis Testing on the Mean

We will consider testing:


\[H_0: \mu = \mu_0 \] \[H_1: \mu \not= \mu_0 \]


Let \(x_1,x_2,\ldots,x_n\) be a random sample for a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\).

If the null hypothesis \(H_0: \mu=\mu_0\) is true, the random variable

\[T=\dfrac{\bar{x}-\mu_0}{\hat{s}/\sqrt{n}}\]

has a \(t\) distribution \(n-1\) degrees of freedom.


13.1.1 Example

A company wants to improve sales. Previous sales data indicate that the mean sale was 220 euros per transaction. After training the sales force, recent sales data were taken from a sample of 9 salesmen. The company needs to know if the training had worked.


Solution


  • Data

\[ x= \{ 203, 229, 215, 220, 223, 233, 208, 228, 209 \} \]

\[n=9; \,\,\,\, \bar{x}=218.67 ; \,\,\,\, s = 10.52.\]

  • Hypothesis

\[H_0: \mu = 220\] \[H_1: \mu \neq 220\]

  • Test statistic

\[T=\dfrac{\bar{x}-\mu}{S/\sqrt{n}} = \dfrac{218.67-220}{10.52/\sqrt{9}}= -0.38 \]

  • Decision


  • Critical Value

We would reject \(H_0\) if \(T\) were less than \(t_\alpha\) or greater than \(t_\alpha\) (determined used a \(t\)-table)

\[\alpha = 0.05; \,\, df = 8\]


The test statistic \(T= \pm 0.38\) is not less than \(t_\alpha= - 2.31\) nor greater than \(t_\alpha= \pm 2.31\). Then, we fail to reject the null hypothesis. That is, the test statistic does not fall in the critical region.
  • p-value

We would determine the area under a \(t_{df}\) curve, to the right of \(T\) and to the left of \(-T\)

\[T = \pm 0.38; \,\, df = 8\]


The \(p\)-value of the test is 0.71, which is greater than the significance level \(\alpha = 0.05\). Then, we fail to reject the null hypothesis. The conclusion is the same regardless of the approach use.


  • Interpretation

There is insufficient evidence, at the \(\alpha= 0.05\) level, to conclude that the mean sale is not different from 220 euros (per transaction).


  • Reporting

A one-sample t-test was computed to determine whether the mean sales was different to the population normal mean sales (220).


Your turn

  • The company thinks that there is a difference (that the mean sales increased after the training). Test this hypothesis.