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Math
Posted 3 months ago

A sociologist studying marriage in the United States and Canada wanted to compare how old, on average, women in each country are when they first get married. The sociologist obtained a random sample of married women from each country. Here is a summary of the ages at first marriage for the women in each sample:
\begin{tabular}{lrr} 
Age at first marriage & US & Canada \\
\hline Mean & 25.5 & 26.4 \\
Standard deviation & 4.0 & 3.5 \\
Number of women & 108 & 102
\end{tabular}

The researcher wants to use these results to test H0:μUSμC=0\mathrm{H}_{0}: \mu_{\mathrm{US}}-\mu_{\mathrm{C}}=0 versus Ha:μUSμC<0\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{US}}-\mu_{\mathrm{C}}<0. Assume that all conditions have been met.

What is the P\mathrm{P}-value associated with these sample results?
Choose 1 answer:
(A) PP-value <0.01<0.01
(B) 0.01P0.01 \leq P-value <0.05<0.05
(C) 0.05P0.05 \leq P-value <0.10<0.10
(D) 0.10P0.10 \leq P-value <0.20<0.20
(E) PP-value 0.20\geq 0.20
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
Calculate the standard error of the difference in sample means
step 2
The standard error (SE) is calculated using the formula SE=sUS2nUS+sC2nCSE = \sqrt{\frac{s_{US}^2}{n_{US}} + \frac{s_{C}^2}{n_{C}}}, where sUSs_{US} and sCs_{C} are the standard deviations for the US and Canada, and nUSn_{US} and nCn_{C} are the sample sizes for the US and Canada, respectively
step 3
Plug in the values: SE=4.02108+3.52102SE = \sqrt{\frac{4.0^2}{108} + \frac{3.5^2}{102}}
step 4
Calculate the test statistic (t-value) using the formula t=(xˉUSxˉC)(μUSμC)SEt = \frac{(\bar{x}_{US} - \bar{x}_{C}) - (\mu_{US} - \mu_{C})}{SE}, where xˉUS\bar{x}_{US} and xˉC\bar{x}_{C} are the sample means for the US and Canada, and (μUSμC)(\mu_{US} - \mu_{C}) is the hypothesized difference in population means under the null hypothesis
step 5
Since the null hypothesis states that (μUSμC)=0(\mu_{US} - \mu_{C}) = 0, the formula simplifies to t=(xˉUSxˉC)SEt = \frac{(\bar{x}_{US} - \bar{x}_{C})}{SE}
step 6
Plug in the values: t=(25.526.4)SEt = \frac{(25.5 - 26.4)}{SE}
step 7
Use the calculated t-value to find the P-value from the t-distribution with degrees of freedom approximately equal to the smaller of nUS1n_{US} - 1 and nC1n_{C} - 1
step 8
Since the alternative hypothesis is μUSμC<0\mu_{US} - \mu_{C} < 0, we are looking for the area to the left of the calculated t-value in the t-distribution
step 9
Determine the range in which the P-value falls based on the t-distribution table or a statistical software
Answer
The P-value associated with these sample results falls within one of the given ranges (A, B, C, D, or E).
Key Concept
Hypothesis testing and P-value determination
Explanation
The P-value in a hypothesis test measures the probability of obtaining a test statistic as extreme as the one observed, under the assumption that the null hypothesis is true. It helps us decide whether to reject the null hypothesis. In this case, we are looking for the probability that the difference in means is less than zero, given the observed sample data.

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