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Chapter 9: Problem 2

Large samples of women and men are obtained, and the hemoglobin level is measured in each subject. Here is the \(95 \%\) confidence interval for the difference between the two population means, where the measures from women correspond to population 1 and the measures from men correspond to population 2 : \(-1.76 \mathrm{~g} / \mathrm{dL}<\mu_{1}-\mu_{2}<-1.62 \mathrm{~g} / \mathrm{dL}\) a. What does the confidence interval suggest about equality of the mean hemoglobin level in women and the mean hemoglobin level in men? b. Write a brief statement that interprets that confidence interval. c. Express the confidence interval with measures from men being population 1 and measures from women being population \(2 .\)

Short Answer

Expert verified
a) Women have lower mean hemoglobin levels than men. b) Mean hemoglobin level in women is between 1.62 and 1.76 g/dL lower than in men with 95% confidence. c) Confidence interval: \(1.62 \mathrm{~g} / \mathrm{dL} < \mu_{2} - \mu_{1} < 1.76 \mathrm{~g} / \mathrm{dL}\).

Step by step solution

01

- Understanding the confidence interval

The confidence interval given is \(-1.76 \mathrm{~g} / \mathrm{dL}<\mu_{1}-\mu_{2}<-1.62 \mathrm{~g} / \mathrm{dL}\), where \(\mu_{1}\) represents the mean hemoglobin level for women and \(\mu_{2}\) for men. This interval tells us the range within which the true difference in means lies with 95% confidence.
02

- Analyzing equality of means (Part a)

Since the entire confidence interval is below 0, it suggests that \(\mu_{1}\) is less than \(\mu_{2}\). Therefore, the mean hemoglobin level in women is significantly lower than that in men.
03

- Interpret the confidence interval (Part b)

The interval \(-1.76 \mathrm{~g} / \mathrm{dL}<\mu_{1}-\mu_{2}<-1.62 \mathrm{~g} / \mathrm{dL}\) implies that we are 95% confident that the true mean hemoglobin level for women is between 1.62 and 1.76 g/dL lower than that for men.
04

- Convert the confidence interval (Part c)

To express the interval with men as population 1 and women as population 2, we switch the roles of \mu_{1}\ and \mu_{2}\. The earlier interval becomes: \(1.62 \mathrm{~g} / \mathrm{dL} < \mu_{2} - \mu_{1} < 1.76 \mathrm{~g} / \mathrm{dL}\).

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Key Concepts

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

population means
When we talk about population means in statistics, we're looking at the average of a particular characteristic within an entire group or population. For instance, if we want to know the average hemoglobin level in women, we consider the entire female population. Here, \(\mu_{1}\) represents the mean hemoglobin level for women, whereas \(\mu_{2}\) does the same for men. By comparing these population means, we can gain insights into differences or similarities across groups. Confidence intervals help by providing a range that likely includes the true difference between these averages.
hemoglobin levels
Hemoglobin is a protein in red blood cells that carries oxygen around the body. Measuring hemoglobin levels is essential for assessing general health and diagnosing medical conditions like anemia. In this exercise, we're using hemoglobin levels to compare two populations: women and men. Generally, hemoglobin levels are measured in grams per deciliter (g/dL). Understanding these measurements is crucial not only for health reasons but also for interpreting statistical differences between groups.
statistical interpretation
Statistical interpretation involves drawing meaningful conclusions from data analyses. In this exercise, the confidence interval provided helps us understand the difference in mean hemoglobin levels between women and men. We interpret the interval \( -1.76 \,\text{g/dL} < \mu_{1} - \mu_{2} < -1.62 \,\text{g/dL} \) to mean we're 95% confident that the true difference in means falls within this range. This interpretation helps create a clear understanding of the observed difference between the two populations.
equality of means
The concept of equality of means examines whether two population averages are statistically the same. In our case, we are checking if the mean hemoglobin levels for women and men are equal. The given confidence interval suggests that \(\mu_{1}\) is consistently lower than \(\mu_{2}\), hence the mean hemoglobin level in women is significantly lower than in men. Since the interval does not include 0, we can conclude that the mean hemoglobin levels for the two populations are not equal.

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Most popular questions from this chapter

Listed below are heights (in.) of mothers and their first daughters. The data are from a journal kept by Francis Galton. (See Data Set 5 "Family Heights" in Appendix B.) Use a \(0.05\) significance level to test the claim that there is no difference in heights between mothers and their first daughters. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|c|} \hline \text { Height of Mother } & 68.0 & 60.0 & 61.0 & 63.5 & 69.0 & 64.0 & 69.0 & 64.0 & 63.5 & 66.0 \\ \hline \text { Height of Daughter } & 68.5 & 60.0 & 63.5 & 67.5 & 68.0 & 65.5 & 69.0 & 68.0 & 64.5 & 63.0 \\ \hline \end{array} $$

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, \(P\) -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Rhinoviruses typically cause common colds. In a test of the effectiveness of echinacea, 40 of the 45 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 88 of the 103 subjects developed rhinovirus infections (based on data from "An Evaluation of Echinacea Angustifolia in Experimental Rhinovirus Infections," by Turner et al., New England Journal of Medicine, Vol. 353, No. 4 ). We want to use a \(0.05\) significance level to test the claim that echinacea has an effect on rhinovirus infections. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, does echinacea appear to have any effect on the infection rate?

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, \(P\) -value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. In a randomized controlled trial in Kenya, insecticide-treated bednets were tested as a way to reduce malaria. Among 343 infants using bednets, 15 developed malaria. Among 294 infants not using bednets, 27 developed malaria (based on data from "Sustainability of Reductions in Malaria Transmission and Infant Mortality in Western Kenya with Use of Insecticide-Treated Bednets," by Lindblade et al., Journal of the American Medical Association, Vol. 291, No. 21 ). We want to use a \(0.01\) significance level to test the claim that the incidence of malaria is lower for infants using bednets. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, do the bednets appear to be effective?

The sample size needed to estimate the difference between two population proportions to within a margin of error \(E\) with a confidence level of \(1-\alpha\) can be found by using the following expression: $$ E=z_{\alpha / 2} \sqrt{\frac{p_{1} q_{1}}{n_{1}}+\frac{p_{2} q_{2}}{n_{2}}} $$ Replace \(n_{1}\) and \(n_{2}\) by \(n\) in the preceding formula (assuming that both samples have the same size) and replace each of \(p_{1}, q_{1}, p_{2}\), and \(q_{2}\) by \(0.5\) (because their values are not known). Solving for \(n\) results in this expression: $$ n=\frac{z_{\alpha / 2}^{2}}{2 E^{2}} $$ Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want \(95 \%\) confidence that your error is no more than \(0.03\).

20\. Does Aspirin Prevent Heart Disease? In a trial designed to test the effectiveness of aspirin in preventing heart disease, 11,037 male physicians were treated with aspirin and 11,034 male physicians were given placebos. Among the subjects in the aspirin treatment group, 139 experienced myocardial infarctions (heart attacks). Among the subjects given placebos, 239 experienced myocardial infarctions (based on data from "Final Report on the Aspirin Component of the Ongoing Physicians' Health Study," New England Journal of Medicine, Vol. 321 : \(129-135)\). Use a \(0.05\) significance level to test the claim that aspirin has no effect on myocardial infarctions. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, does aspirin appear to be effective?In a trial designed to test the effectiveness of aspirin in preventing heart disease, 11,037 male physicians were treated with aspirin and 11,034 male physicians were given placebos. Among the subjects in the aspirin treatment group, 139 experienced myocardial infarctions (heart attacks). Among the subjects given placebos, 239 experienced myocardial infarctions (based on data from "Final Report on the Aspirin Component of the Ongoing Physicians' Health Study," New England Journal of Medicine, Vol. 321 : \(129-135)\). Use a \(0.05\) significance level to test the claim that aspirin has no effect on myocardial infarctions. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, does aspirin appear to be effective?
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