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Chapter 9: Problem 28

Suppose that 935 smokers each received a nicotine patch, which delivers nicotine to the bloodstream at a much slower rate than cigarettes do. Dosage was decreased to 0 over a 12 -week period. Of these 935 people, 245 were still not smoking 6 months after treatment. Assume this sample is representative of all smokers. a. Use the given information to estimate the proportion of all smokers who, when given this treatment, would refrain from smoking for at least 6 months. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

Short Answer

Expert verified
a. The estimate for the proportion who would refrain from smoking for at least 6 months given the treatment is approximately 0.262 or 26.2%. b. The conditions for the margin of error to be calculated are met. c. The margin of error, at a 95% confidence level, is approximately 0.029 or 2.9%. d. This suggests that the true population proportion who would quit smoking for at least 6 months when given this treatment is likely to fall within the range of 23.3% to 29.1%.

Step by step solution

01

Calculate Proportion

The proportion of smokers who, given the treatment, refrain from smoking for at least six months can be estimated by dividing the number of successes (people who quit smoking) by the total number of trials (total people who were treated). In this case, there were \(245\) successful outcomes out of a total of \(935\) trials. So the estimated proportion \(p\) is \(\frac{245}{935} \approx 0.262\).
02

Verify Conditions for Margin of Error

To check if the conditions for the margin of error formula are met, we need to ensure that both \(n \cdot p \geq 10\) and \(n \cdot (1 - p) \geq 10\), where \(n\) is the sample size and \(p\) is the proportion. In this case, \(n = 935\), and \(p = 0.262\), so both conditions are met.
03

Calculate Margin of Error

The standard error of a proportion is defined as \(SE = \sqrt{ \frac{p(1 - p)}{n}}\), where \(p\) is the proportion and \(n\) is the sample size. Plugging in the values from the problem, we find that \(SE \approx 0.015\). The margin of error at a 95% confidence level for a large sample size is given by \(ME = 1.96 \cdot SE\). Plugging in our standard error from before we get \(ME = 1.96 \cdot 0.015 \approx 0.029\). This is our margin of error.
04

Interpret Margin of Error

The margin of error is the range that the true population proportion is likely to fall within (with a certain level of confidence, in this case 95%). In this context, the margin of error suggests that the true proportion of all smokers who would successfully quit smoking for at least 6 months, when given this treatment, could be as low as \(26.2% - 2.9% = 23.3%\), or as high as \(26.2% + 2.9% = 29.1%\), with 95% confidence.

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Most popular questions from this chapter

Appropriate use of the interval $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(\hat{p}\), indicate whether the sample size is large enough for this interval to be appropriate. a. \(n=50\) and \(\hat{p}=0.30\) b. \(n=50\) and \(\hat{p}=0.05\) c. \(n=15\) and \(\hat{p}=0.45\) d. \(n=100\) and \(\hat{p}=0.01\)

A random sample will be selected from the population of all students enrolled at a large college. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all students who use public transportation to travel to campus. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) i. \(\quad n=300, p=0.3\) ii. \(\quad n=700, p=0.2\) iii. \(n=1,000, p=0.1\)

A researcher wants to estimate the proportion of students enrolled at a university who eat fast food more than three times in a typical week. Would the standard error of the sample proportion \(\hat{p}\) be smaller for random samples of size \(n=50\) or random samples of size \(n=200 ?\)

Suppose that county planners are interested in learning about the proportion of county residents who would pay a fee for a curbside recycling service if the county were to offer this service. Two different people independently selected random samples of county residents and used their sample data to construct the following confidence intervals for the proportion who would pay for curbside recycling: Interval 1:(0.68,0.74) Interval 2:(0.68,0.72) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals are associated with a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?
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