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Listed below are the numbers of words spoken in a day by each member of eight different randomly selected couples. Complete parts (a) and (b) below.

\begin{tabular}{lcccccccc}
\hline
Male & 16,275 & 25,989 & 1,418 & 7,921 & 19,192 & 15,904 & 14,323 & 26,696 \\
\hline
Female & 23,914 & 12,888 & 17,819 & 17,515 & 13,425 & 17,291 & 15,738 & 18,182 \\
\hline
\end{tabular}

(a) What is the conclusion based on the hypothesis test?

Since the [tex]$P$[/tex]-value is greater than the significance level, fail to reject the null hypothesis. There is not sufficient evidence to support the claim that males speak fewer words in a day than females.

(b) Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

The confidence interval is [tex]$\square$[/tex] word(s) [tex]$\ \textless \ \mu_d\ \textless \ $[/tex] [tex]$\square$[/tex] word(s).

(Round to the nearest integer as needed.)



Answer :

GinnyAnswer
Let's walk through the detailed steps necessary to solve the problem you have presented:

### Step-by-Step Solution

### Part (a): Hypothesis Test Conclusion

Given the hypothesis test conclusion which states:

"Since the [tex]$P$[/tex]-value is greater than the significance level, fail to reject the null hypothesis. There is not sufficient evidence to support the claim that males speak fewer words in a day than females."

This conclusion means that the calculated [tex]$P$[/tex]-value, which helps determine the statistical significance of our results, is higher than the chosen threshold for significance (often denoted as [tex]$\alpha$[/tex]). When the [tex]$P$[/tex]-value is greater than [tex]$\alpha$[/tex], we do not have enough evidence to reject the null hypothesis.

In our context, the null hypothesis (H0) is likely that there is no difference in the number of words spoken by males and females in a day. The alternative hypothesis (H1) would be that males speak fewer words in a day than females. Because the null hypothesis is not rejected, we conclude there is insufficient evidence to support that males speak fewer words than females.

### Part (b): Constructing the Confidence Interval

To answer part (b), we need to construct a confidence interval for the difference in the mean number of words spoken by males and females. The result provides the following statistics:
- Mean difference ([tex]$\overline{d}$[/tex]): -1131.75 words
- Standard deviation of differences (sd): 9931.58 words
- Standard error (SE): 3511.34 words
- Critical value (from the t-distribution for a 95% confidence level): 2.3646
- Margin of error (ME): 8303.01 words

To construct the confidence interval for the mean difference, we use the formula:

[tex]\[ \text{Confidence Interval} = (\overline{d} - \text{ME}, \overline{d} + \text{ME}) \][/tex]

Substituting the given values:

[tex]\[ \text{Lower bound} = -1131.75 - 8303.01 = -9434.76 \][/tex]
[tex]\[ \text{Upper bound} = -1131.75 + 8303.01 = 7171.26 \][/tex]

Thus, the confidence interval (rounded to the nearest integer) is:

[tex]\[ -9435 \text{ words} < \mu_d < 7171 \text{ words} \][/tex]

### Conclusion: Confidence Interval

The feature of the confidence interval that leads to the same conclusion reached in part (a) is that the interval contains 0. When 0 is within the confidence interval, it indicates that there is no statistically significant difference between the means of the two groups (males and females, in this case). This aligns with the hypothesis test conclusion that there is not sufficient evidence to suggest a difference in the number of words spoken by males and females.

Thus, the confidence interval is:

[tex]\[ -9435 \text{ words} < \mu_d < 7171 \text{ words} \][/tex]

The presence of 0 within this interval explains why we fail to reject the null hypothesis in the hypothesis test.

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