Chapter 10, Problem 10.1TI

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.

	Sample Size	Sample Mean	Sample Standard Deviation
Population A	25	5	1
Population B	16	4.7	1.2

Table 10.2

NOTE When the sum of the sample sizes is larger than 30 ( $n_{\text{1}} +n_{\text{2}}>\text{3}0$ ) you can use the normal distribution to approximate the Student's t.

To determine

Whether any difference in the means of the given two samples.

Answer to Problem 10.1TI

There is no sufficient evidence to conclude that the means of the two populations are not the same.

Explanation of Solution

Given information:

The given table shows two samples:

	Sample size	Sample mean	Sample standard deviation
Population A	25	5	1
Population B	16	4.7	1.2

Both have normal distribution. The means of the two populations are thought to be same. Test at the 5% level of significance.

Solution:

The null hypothesis is $\begin{array}{l}{H}_0:{\mu }_a={\mu }_b/\\ H_0:{\mu }_a-{\mu }_b=0\end{array}$

The alternative hypothesis is:

$\begin{array}{l}{H}_a:{\mu }_a\ne {\mu }_b/\\ H_a:{\mu }_a-{\mu }_b\ne 0\end{array}$

The significant level, $\alpha =5\%=0.05$

The p-value is 0.4125, which is much higher than 0.05, so we decline to reject the null hypothesis.

Hence, there is no sufficient evidence to conclude that the means of the two populations are not the same.