Question

Two samples are shown in Table $10.2$. Both have normal distributions. The means for the two populations are thought to be the same. Is there a difference in the means? Test at the $5\%$ level of significance.

Short answer

There isn't enough data to establish that the two populations' means aren't the same.

Step-by-step solution

Given information

Given a sample of two populations and the level of significance, $a=5\%$

Explanation

The null hypothesis is:

$$\begin{gathered} H_0:{\mu }_a={\mu }_{b}I \\ {H}_0:{\mu }_a-{\mu }_b=0 \end{gathered}$$

The alternative hypothesis is:

$$\begin{gathered} H_a:{\mu }_a\ne {\mu }_{b}I \\ {H}_a:{\mu }_a-{\mu }_b\ne 0 \end{gathered}$$

The significant level,

$$\begin{gathered} a=5\% \\ a=0.05 \end{gathered}$$

We decline to reject the null hypothesis because the p-value is $0.4125$, which is significantly greater than$0.05$. As a result, there is no reason to believe that the mean of the two populations differs.