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Chapter 10: Problem 4

A National Center for Health Statistics data brief published in 2015 (Nr. 181) looked at the association between lung obstruction and smoking status in adults 40 to 79 years old. In a random sample of 6927 adults without any lung obstruction, \(54.1 \%\) never smoked. In a random sample of 1146 adults with lung obstruction (such as asthma or COPD), \(23.8 \%\) never smoked. a. Find and interpret a point estimate of the difference between the proportion of adults without and with lung obstruction who never smoked. b. A \(99 \%\) confidence interval for the true difference is \((0.267,0.339) .\) Interpret. c. What assumptions must you make for the interval in part b to be valid?

Short Answer

Expert verified
a. The point estimate is 0.303. b. We are 99% confident that the true difference is between 0.267 and 0.339. c. Assume large, random, and independent samples; success-failure condition met.

Step by step solution

01

Find Proportion of Non-Smokers without Lung Obstruction

We are given that 54.1% of 6927 adults without lung obstruction never smoked. We need to find the proportion in decimal form.\[ p_1 = \frac{54.1}{100} = 0.541 \]
02

Calculate Number of Non-Smokers without Lung Obstruction

To find the number of non-smokers among the 6927 adults without lung obstruction, multiply the total number by the proportion.\[ n_1 \times p_1 = 6927 \times 0.541 = 3748.107 \]
03

Find Proportion of Non-Smokers with Lung Obstruction

We are given that 23.8% of 1146 adults with lung obstruction never smoked. Convert this percentage to a decimal.\[ p_2 = \frac{23.8}{100} = 0.238 \]
04

Calculate Number of Non-Smokers with Lung Obstruction

To find the number of non-smokers among the 1146 adults with lung obstruction, multiply the total number by the proportion.\[ n_2 \times p_2 = 1146 \times 0.238 = 272.748 \]
05

Calculate Point Estimate of the Difference in Proportions

The point estimate for the difference between the proportions is the difference between \( p_1 \) and \( p_2 \).\[ \hat{p_1} - \hat{p_2} = 0.541 - 0.238 = 0.303 \]
06

Interpret Point Estimate

The point estimate 0.303 indicates that the proportion of adults without lung obstruction who never smoked is 30.3% higher than the proportion of adults with lung obstruction who never smoked.
07

Interpret 99% Confidence Interval

The 99% confidence interval for the difference in proportions is (0.267, 0.339). This means we are 99% confident that the true difference in proportions of adults who never smoked between those without and with lung obstruction lies between 26.7% and 33.9%.
08

Identify Assumptions for Validity

We're assuming the sample proportions are normally distributed, which is valid if the sample size is large. Both samples should be random and independent, and the success-failure condition must be met: \( np \geq 5 \) and \( n(1-p) \geq 5 \) for both groups.

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Key Concepts

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

Confidence Interval
A confidence interval offers a range of values that is likely to contain the true difference between two population parameters, such as proportions in this case. This range provides an estimate of the "margin of error" associated with a sample statistic. When we have a 99% confidence interval, it suggests we are 99% certain that the true parameter lies within this interval.
The interval (0.267, 0.339) indicates that the true difference in the proportion of adults who never smoked, between those without and with lung obstruction, falls in this range. This means that the non-smoking rate for adults without lung obstruction likely exceeds that of adults with lung obstruction by at least 26.7% and at most 33.9%.
  • Confidence interval helps in understanding the reliability of the estimate.
  • A wider interval may suggest more uncertainty about the estimate.
  • Since we use a 99% confidence interval, it is quite a robust estimate.
Point Estimate
The point estimate is a single value estimate representing the population parameter, often extracted directly from sample data. In terms of difference in proportions, it shows the difference in rates or percentages between two groups.
In this exercise, the point estimate obtained is 0.303 or 30.3%. This result shows the estimated proportion of adults without lung obstruction who never smoked is 30.3% higher compared to those with lung obstruction.
Understanding point estimates involves
  • Calculating a straightforward comparison between two proportions.
  • Viewing it as a specific value that falls within the broader confidence interval.
  • Recognizing it as an unbiased estimate of the true population difference.
Point estimates provide a useful snapshot, but should always be considered alongside confidence intervals for a more complete picture.
Proportions
Proportions form the basis of many statistical analyses, especially when comparing two or more groups. Proportions express part of the whole and can be converted from or to percentages. They help in understanding the comparison of characteristics or behaviors within different groups.
In this exercise, the proportions tell us how many adults in both groups (with and without lung obstruction) have never smoked, expressed relative to the entire sample of each group.
  • For adults without lung obstruction, 54.1% never smoked.
  • For adults with lung obstruction, 23.8% never smoked.
These proportions highlight the differences in smoking behavior between the two groups. Calculating proportions accurately is crucial, and using them allows us to make informed inferences about wider populations.

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Most popular questions from this chapter

Steve Solomon, the owner of Leonardo's Italian restaurant, wonders whether a redesigned menu will increase, on the average, the amount that customers spend in the restaurant. For the following scenarios, pick a statistical method from this chapter that would be appropriate for analyzing the data, indicating whether the samples are independent or dependent, which parameter is relevant, and what inference method you would use: a. Solomon records the mean sales the week before the change and the week after the change and then wonders whether the difference is "statistically significant." b. Solomon randomly samples 100 people and shows them each both menus, asking them to give a rating between 0 and 10 for each menu. c. Solomon randomly samples 100 people and shows them each both menus, asking them to give an overall rating of positive or negative to each menu. d. Solomon randomly samples 100 people and randomly separates them into two groups of 50 each. He asks those in Group 1 to give a rating to the old menu and those in Group 2 to give a rating to the new menu, using a 0 to 10 rating scale.

A study \(^{14}\) compared personality characteristics between 49 children of alcoholics and a control group of 49 children of nonalcoholics who were matched on age and gender. On a measure of well-being, the 49 children of alcoholics had a mean of \(26.1(s=7.2)\) and the 49 subjects in the control group had a mean of \(28.8(s=6.4) .\) The difference scores between the matched subjects from the two groups had a mean of \(2.7(s=9.7)\). a. Are the groups to be compared independent samples or dependent samples? Why? b. Show all steps of a test of equality of the two population means for a two- sided alternative hypothesis. Report the P-value and interpret. c. What assumptions must you make for the inference in part b to be valid?

Normal assumption The methods of this section make the assumption of a normal population distribution. Why do you think this is more relevant for small samples than for large samples? (Hint: What shape does the sampling distribution of \(\bar{x}_{1}-\bar{x}_{2}\) have for large samples, regardless of the actual shape of the population distributions?)

Two new programs were recently proposed at the University of Florida for treating students who suffer from math anxiety. Program A provides counseling sessions, one session a week for six weeks. Program \(\mathrm{B}\) supplements the counseling sessions with short quizzes that are designed to improve student confidence. For ten students suffering from math anxiety, five were randomly assigned to each program. Before and after the program, math anxiety was measured by a questionnaire with 20 questions relating to different aspects of taking a math course that could cause anxiety. The study measured, for each student, the drop in the number of items that caused anxiety. The sample values were Program A: 0,4,4,6,6 Program B: 6,12,12,14,16 Using software, analyze these data. Write a report, summarizing the analyses and interpretations.

The Women's Health Initiative conducted a randomized experiment to see whether hormone therapy was helpful for postmenopausal women. The women were randomly assigned to receive the estrogen plus progestin hormone therapy or a placebo. After five years, 107 of the 8506 on the hormone therapy developed cancer and 88 of the 8102 in the placebo group developed cancer. Is this a significant difference? a. Set up notation and state assumptions and hypotheses. b. Find the test statistic and P-value and interpret. (If you prefer, use software, such as MINITAB or a web app, for which you can conduct the analysis by entering summary counts.) c. What is your conclusion for a significance level of \(0.05 ?\) (The study was planned to be eight years long but was stopped after five years because of increased heart and cancer problems for the therapy group. This shows a benefit of doing two-sided tests, because results sometimes have the opposite direction from the expected one.)
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