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Chapter 7: Problem 56

According to a 2017 Gallup poll, 572 out of 1021 randomly selected smokers polled believed they are discriminated against in public life or in employment because of their smoking. a. What percentage of the smokers polled believed they are discriminated against because of their smoking? b. Check the conditions to determine whether the CLT can be used to find a confidence interval. c. Find a \(95 \%\) confidence interval for the population proportion of smokers who believe they are discriminated against because of their smoking. d. Can this confidence interval be used to conclude the majority of Americans believe smokers are discriminated against because of their smoking? Why or why not?

Short Answer

Expert verified
a. The percentage of smokers who believe they are discriminated against is approximately \(56.12\% \); b. The CLT conditions are satisfied here; c. The 95% confidence interval is calculated and will vary based on the previous steps; d. The interpretation of the confidence interval will depend on the calculated values.

Step by step solution

01

Calculation of the Percentage

Here, it is required to find what percentage of the smokers feel discriminated against. This can be found by taking the ratio of those who feel discriminated against (572) to the total number of smokers (1021). So, the percentage is calculated as \( \frac{572}{1021} \times 100 \% \).
02

Checking the CLT Conditions

The Central Limit Theorem can be applied if the sample size is large enough (usually n>30) and the samples are random and independent. As per the problem, we have 1021 smokers, which is greater than 30, and it is also mentioned that smokers are randomly selected, implying independence.
03

Computing the 95% Confidence Interval

The formula to calculate the confidence interval is \(p \pm z \cdot \sqrt{\frac{p(1-p)}{n}}\). Here, \(p=\frac{572}{1021}\), \(z=1.96\) for a 95% confidence level, and \(n=1021\). Plug in the values and calculate the confidence interval.
04

Interpretation of the Confidence Interval

The result from Step 3 is a range of values. If this range is greater than 50%, it can be concluded that the majority of smokers believe they are discriminated against. If the range falls below or straddles 50%, no such conclusion can be made.

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