Chapter 10.2, Problem 53E

(a)

To determine

To find: The average number of words being spoken in a day by the male and female students at the university.

Answer to Problem 53E

The decision is “fail reject the null hypothesis”.

Explanation of Solution

Given information:

Researchers equipped random samples of 56 male and 56 female students from a large university with a small device that secretly records sound for a random 30 seconds during each 12.5-minute period over two days.

Formula used:

  $t=\frac{\overline{x_1}-\overline{x_2}}{\sqrt{\frac{x_1^2}{n_1}+\frac{x_2^2}{n_2}}}$

Calculation:

Here, the two independent samples are measured. Hence, two sample t test is used.

The hypotheses are given below:

Null hypothesis:

  $H_0:{\mu }_1-{\mu }_2=0$

Alternative hypothesis:

  $H_a:{\mu }_1-{\mu }_2\ne 0$

The test statistic is 0.25, which is obtained below:

  $\begin{array}{l}t=\frac{\overline{x_1}-\overline{x_2}}{\sqrt{\frac{x_1^2}{n_1}+\frac{x_2^2}{n_2}}}\\ =\frac{16569-16177}{\sqrt{\frac{{9108}^2}{56}+\frac{{7520}^2}{56}}}\\ t=0.25\end{array}$

The degrees of freedom is,

  $\begin{array}{l}df=smallerof\left(n_1-1\right)\left(n_2-1\right)\\ =56-1\\ df=55\end{array}$

P-value:

The P-value is obtained Excel function “=TDIST(x, degrees of freedom, tails)”.

Output obtained using Excel is as follows:

  

From the output, the P-value is 0.804.

 Decision rule:

If the P-value of the test is less than the level of significance, reject. Otherwise, fail to reject the null hypothesis.

Conclusion:

The level of significance is 0.05 and the P-value is 0.804.

Here, P-value is greater than the level of significance.

Thus, the decision is “fail reject the null hypothesis”.

(b)

To determine

To analyse: The interpretation of the P-value

Answer to Problem 53E

There is no evidence to conclude that the difference in the average number of words spoken in a day by male and female students at this university.

Explanation of Solution

Given information:

Researchers equipped random samples of 56 male and 56 female students from a large university with a small device that secretly records sound for a random 30 seconds during each 12.5-minute period over two days.

There is no evidence to conclude that the difference in the average number of words spoken in a day by male and female students at this university.

Conclusion:

There is no evidence to conclude that the difference in the average number of words spoken in a day by male and female students at this university.