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The number of words spoken in a day by each member of 6 couples is recorded.

It is claimed that males speak fewer than females.

Corresponding to the given claim, the following hypotheses are set up:

Null Hypothesis: The mean of the difference between the number of words spoken by males and females is equal to 0.

\({H_0}:{\mu _d} = 0\)

Alternative Hypothesis: The mean difference between the number of words spoken by males and females is less than 0.

\({H_1}:{\mu _d} < 0\)

The test is left-tailed.

The following table shows the differences in the number of words spoken by males and females for each matched pair:

The number of pairs (n) is equal to 8.

The mean value of the differences is computed below:

\(\begin{array}{c}\bar d = \frac{{\left( { - 8941} \right) + 13032 + ...... + 7168}}{8}\\ = - 1678\end{array}\)

The standard deviation of the differences is computed below:

\(\begin{array}{c}{s_d} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({d_i} - \bar d)}^2}} }}{{n - 1}}} \\ = \sqrt {\frac{{{{\left( { - 8941 - \left( { - 1678} \right)} \right)}^2} + {{\left( {13032 - \left( { - 1678} \right)} \right)}^2} + ....... + {{\left( {7168 - \left( { - 1678} \right)} \right)}^2}}}{{8 - 1}}} \\ = 10052.87\end{array}\)

The mean value of the differences for the population of matched pairs \(\left( {{\mu _d}} \right)\) is considered to be equal to 0.

The value of the test statistic is computed as shown:

\(\begin{array}{c}t = \frac{{\bar d - {\mu _d}}}{{\frac{{{s_d}}}{{\sqrt n }}}}\\ = \frac{{ - 1678 - 0}}{{\frac{{10052.87}}{{\sqrt 8 }}}}\\ = - 0.472\end{array}\)

The degrees of freedom are computed below:

\(\begin{array}{c}df = n - 1\\ = 8 - 1\\ = 7\end{array}\)

Referring to the t-distribution table, the critical value of t at\(\alpha = 0.05\)and degrees of freedom equal to 7 for a left-tailed test is equal to -1.8946.

The left-tailed p-value for t equal to -0.472 is equal to 0.3256.

a.

Since the test statistic value is greater than the critical value and the p-value is greater than 0.05, the null hypothesis is failed to reject.

There is not enough evidence to support the claim that males speak fewer words in a day than females.

b.

The confidence interval has the following expression:

\(CI = \bar d - E < {\mu _d} < \bar d + E\)

The value of the margin of error (E) is given by the formula:

\(E = {t_{\frac{\alpha }{2}}}\frac{{{s_d}}}{{\sqrt n }}\)

The confidence level to construct the confidence interval for a one-tailed test\(\alpha = 0.05\)is equal to 90%.

Thus, the value of\(\alpha \)for constructing the confidence interval is equal to 0.

The value of\({t_{\frac{\alpha }{2}}}\)for 7 degrees of freedom when\(\alpha = 0.10\)is equal to 1.8946.

Substitute the respective values to compute the margin of error.

\(\begin{array}{c}E = 1.8946 \times \frac{{10052.87}}{{\sqrt 8 }}\\ = 6733.84\end{array}\)

Substitute the required values in the formula to compute the confidence interval.

\(\begin{array}{c} - 1678 - 6733.84 < {\mu _d} < - 1678 + 6733.84\\ - 8412 < {\mu _d} < 5056\end{array}\)

It can be observed that 0 lies within the interval. This implies that the difference in the number of words spoken in a day by men and women can be equal to 0.

Thus it can be concluded that there is not enough evidence to support the claim that males speak fewer words in a day than females.