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The National Health Statistics Reports dated Oct. \(22 .\) 2008 , included the following information on the heights (in.) for non-Hispanic white females: \begin{tabular}{lccc} Age & Sample Size & Sample Mean & Std. Error Mean \\ \hline \(20-39\) & 866 & \(64.9\) & \(.09\) \\ 60 and older & 934 & \(63.1\) & \(.11\) \\ \hline \end{tabular} a. Calculate and interpret a confidence interval at confidence level approximately \(95 \%\) for the difference between population mean height for the younger women and that for the older women. b. Let \(\mu_{1}\) denote the population mean height for those aged 20-39 and \(\mu_{2}\) denote the population mean height for those aged 60 and older. Interpret the hypotheses \(H_{0}: \mu_{1}-\mu_{2}=1\) and \(H_{a}: \mu_{1}-\mu_{2}>1\), c. Based on the \(P\)-value calculated in (b) would you reject the null hypothesis at any reasonable significance level? Explain your reasoning. d. What hypotheses would be appropriate if \(\mu_{1}\) referred to the older age group, \(\mu_{2}\) to the younger age group, and you wanted to see if there was compelling evidence for concluding that the population mean height for younger women exceeded that for older women by more than 1 in.?

Step by step solution

01

Calculate the Standard Error for the Difference in Means

To calculate the standard error for the difference in means, use the formula:\[SE_{diff} = \sqrt{(SE_1)^2 + (SE_2)^2}\]where \(SE_1 = 0.09\) and \(SE_2 = 0.11\). Thus,\[SE_{diff} = \sqrt{(0.09)^2 + (0.11)^2} = \sqrt{0.0081 + 0.0121} = \sqrt{0.0202} \approx 0.142\]

02

Find the 95% Confidence Interval for the Difference

The 95% confidence interval is given by:\[(\overline{X}_1 - \overline{X}_2) \pm z^* \cdot SE_{diff}\]where \(\overline{X}_1 = 64.9\), \(\overline{X}_2 = 63.1\), and \(z^*\) for 95% confidence is approximately 1.96. Thus,\[64.9 - 63.1 \pm 1.96 \times 0.142 = 1.8 \pm 0.278\]This results in the interval \([1.522, 2.078]\). This means we are 95% confident that the actual difference in population means falls within this interval.

03

Interpret Hypotheses for Given Conditions

\(H_0: \mu_1 - \mu_2 = 1\) means the difference in population means is hypothesized to be exactly 1 inch. \(H_a: \mu_1 - \mu_2 > 1\) means we believe the difference is greater than 1 inch. We test to see if this greater difference is supported by data.

04

P-Value Calculation for Hypothesis Test

For \(H_a: \mu_1 - \mu_2 > 1\), the test statistic is:\[Z = \frac{(\overline{X}_1 - \overline{X}_2) - 1}{SE_{diff}} = \frac{1.8 - 1}{0.142} \approx 5.63\]A Z-score of 5.63 corresponds to a very small P-value (less than 0.05), leading us to reject \(H_0\) at any reasonable significance level.

05

Formulate Hypotheses with Reversed Roles

If \(\mu_1\) refers to the older group and \(\mu_2\) to the younger group, and we want to show that the younger group is taller by more than 1 inch, we set:\(H_0: \mu_2 - \mu_1 = 1\) \(H_a: \mu_2 - \mu_1 > 1\). These hypotheses align with testing if younger women are on average more than 1 inch taller than older women.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Error

The standard error is an important statistic that provides insight into how much the sample mean is expected to vary from the true population mean. It's like a measure of the precision of our sample mean. In the context of the exercise, the standard error helps us understand the variability in the height measurements of women in different age groups.

When calculating the standard error for the difference in means, we use the formula: \[SE_{diff} = \sqrt{(SE_1)^2 + (SE_2)^2}\]This formula considers the standard errors of both groups involved in the comparison. It accounts for the combined uncertainty in measuring each group's mean height.

Smaller standard errors suggest that our sample mean is a good estimate of the population mean. In this exercise, a calculated standard error of about 0.142 indicates relatively low variability in mean height differences, making our results fairly reliable.

Hypothesis Testing

Hypothesis testing is all about making informed decisions based on data. It provides a structured way to evaluate claims or hypotheses about population parameters, such as means or proportions.

In the exercise, our hypotheses were: \(H_0: \mu_1 - \mu_2 = 1\) and \(H_a: \mu_1 - \mu_2 >1\). - **Null Hypothesis \(H_0\)**: This is the assumption that the null (no effect) condition is true. Here, it means that the difference in population mean heights is exactly 1 inch.
- **Alternative Hypothesis \(H_a\)**: This suggests that there is an effect, specifically that the average height difference is greater than 1 inch.

We aim to gather evidence to either support or reject the null hypothesis. If the data show a likelihood that the alternative hypothesis is true, we may reject \(H_0\). This structured testing ensures our conclusions are data-driven rather than based on assumptions.

P-value Interpretation

The P-value is a crucial component in hypothesis testing, as it helps us make decisions about the null hypothesis. It measures the strength of evidence against the null hypothesis, giving us an indication of whether our observed data could occur by random chance.

In the context of the exercise, we calculated a Z-score to determine the likelihood that our observed difference (1.8 inches) could happen if the null hypothesis (\(\mu_1 - \mu_2 = 1\)) were true. A Z-score of 5.63 produced a very small P-value, often less than 0.05, an arbitrary threshold used for significance.

If the P-value is small, it suggests that the observed result is unlikely under the null hypothesis, leading us to reject \(H_0\). In simple terms, a small P-value gives us confidence that there鈥檚 a statistically significant difference greater than the hypothesized 1 inch.

Population Mean Difference

Understanding the concept of population mean difference is vital when comparing two groups. This value represents the average gap between the population means of these groups. In research, it helps identify whether different populations are experiencing measurable changes or differences.

In our exercise, the population mean difference between younger and older non-Hispanic white females is crucial because it helps determine if age affects average height. Here's what this tells us:

• A calculated difference of 1.8 inches suggests, on average, that younger women might be taller than older women.
• Based on statistical analysis, this difference is statistically significant.

Understanding the population mean difference helps in interpreting real-world implications, such as policy-making or health guidelines. In this study, the confidence interval gives further assurance by confirming that the true mean difference likely falls between 1.522 and 2.078 inches, reinforcing the idea that younger women could be taller than their older counterparts.