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Chapter 22: Problem 25

The Harris Poll asked a sample of smokers, "Do you believe that smoking will probably shorten your life, or not?" Of the 1010 people in the sample, 848 said Yes. (a) Harris called residential telephone numbers at random in an attempt to contact an SRS of smokers. Based on what you know about national sample surveys, what is likely to be the biggest weakness in the survey? (b) We will nonetheless act as if the people interviewed are an SRS of smokers. Give a \(95 \%\) confidence interval for the percent of smokers who agree that smoking will probably shorten their lives.

Short Answer

Expert verified
(a) Coverage bias is likely the biggest weakness. (b) The confidence interval is approximately 81.7% to 86.2%.

Step by step solution

01

Identify the Survey's Weakness

Surveys that call residential telephone numbers can have coverage bias. Not all smokers may have landline telephones, and younger people might primarily use mobile phones. This can lead to a non-representative sample.
02

Define Parameters and Calculations

We have 848 smokers who agree that smoking shortens lives out of 1010 respondents. Let \( p \) represent the proportion of smokers who say 'Yes'. Thus, \( \hat{p} = \frac{848}{1010} \approx 0.8396 \).
03

Calculate Standard Error

The standard error (SE) for \( \hat{p} \) is given by \( \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \). Substituting the values: \[ SE = \sqrt{\frac{0.8396 \times (1 - 0.8396)}{1010}} \approx 0.0115. \]
04

Determine the Confidence Interval

For a 95% confidence interval, use a critical value of approximately 1.96. The confidence interval is: \[ \hat{p} \pm 1.96 \times SE. \] Substituting the values: \( 0.8396 \pm 1.96 \times 0.0115 \).
05

Calculate the Confidence Interval Bounds

Calculate the bounds: Lower Bound = \( 0.8396 - 1.96 \times 0.0115 \approx 0.8169 \); Upper Bound = \( 0.8396 + 1.96 \times 0.0115 \approx 0.8623 \).
06

Interpret the Confidence Interval

The 95% confidence interval for the proportion of smokers who believe smoking will shorten their life is approximately 81.7% to 86.2%.

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Key Concepts

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

Survey Bias
When conducting a survey, it's crucial to understand potential biases that may affect the results. Survey bias can lead to non-representative samples and skewed conclusions. In the case of the Harris Poll, one likely source of bias is the reliance on residential phone numbers. Coverage bias occurs when certain groups are underrepresented in the sample. For example, younger people and those without landline phones may not have been reached.
This is problematic because:
  • Not everyone uses or has access to a landline.
  • Younger populations are more likely to only use mobile phones, possibly excluding their views.
Without addressing this bias, the survey results can misrepresent the actual beliefs of the intended population. Therefore, it's important to consider alternative methods of reaching participants, such as incorporating mobile numbers or online surveys.
Simple Random Sample (SRS)
A Simple Random Sample (SRS) is a cornerstone of good survey practice, ensuring that every individual in the population has an equal chance of being selected. This is crucial for obtaining accurate and generalizable results. In the Harris Poll, the goal was to reach an SRS of smokers.
Here's why SRS is important:
  • It reduces bias, making the sample more representative of the population.
  • The results can be generalized to the larger population if the sample is truly random.
However, achieving an SRS may be complex without randomness in participant selection. To maintain the integrity of an SRS, survey designers must ensure that no subset of the population is systematically excluded from selection.
Standard Error
The Standard Error (SE) is a vital concept in statistics, providing a measure of the accuracy of a sample mean by calculating the variability between sample means if the survey were repeated multiple times. In simpler terms, it estimates how much a sample mean is expected to vary from the true population mean.
For the Harris Poll:
  • The formula for SE is \( \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \), where \( \hat{p} \) is the sample proportion and \( n \) is the sample size.
  • For this survey, the SE was found to be approximately 0.0115, indicating a relatively small deviation from the true population mean.
Understanding SE helps in constructing confidence intervals. A smaller SE indicates more reliability in the sample estimate, guiding informed decisions based on the survey results.

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Most popular questions from this chapter

In the National AIDS Behavioral Surveys sample of 2673 adult heterosexuals, \(0.2 \%\) (that's \(0.002\) as a decimal fraction) had both received a blood transfusion and had a sexual partner from a group at high risk of AIDS. Explain why we can't use the large-sample confidence interval to estimate the proportion \(p\) in the population who share these two risk factors.

The value of the \(z\) statistic for the Exercise \(22.22\) is \(2.53\). This test is (a) not significant at either \(\alpha=0.05\) or \(\alpha=0.01\). (b) significant at \(\alpha=0.05\) but not at \(\alpha=0.01\). (c) significant at both \(\alpha=0.05\) and \(\alpha=0.01\).

Students are reluctant to report cheating by other students. A student project put this question to an SRS of 172 undergraduates at a large university: "You witness two students cheating on a quiz. Do you go to the professor?" Only 19 answered Yes. \({ }^{20}\) Give a \(95 \%\) confidence interval for the proportion of all undergraduates at this university who would report cheating.

\(\hat{p}\). Greenville County, South Carolina, has 461,299 adult residents, of which 59,969 are 65 years or older. A survey wants to contact \(n=689\) residents. \({ }^{5}\) (a) Find \(p\), the proportion of Greenville county adult residents who are 65 years or older. (b) If repeated simple random samples of 689 residents are taken, what would be the range of the sample proportion of adults over 65 in the sample according to the 95 part of the 68-95-99.7 rule? (c) The actual survey contacted 689 adults using random digit dialing of residential numbers using a database of exchanges, with no cell phone numbers contacted. The 689 respondents represent a response rate of approximately \(30 \%\). In the sample obtained, 253 of the 689 adults contacted were over 65 . Do you have any concerns treating this as a simple random sample from the population of adult residents of Greenville County? Explain briefly.

Explain whether we can use the \(z\) test for a proportion in these situations. (a) You toss a coin 10 times in order to test the hypothesis \(H_{0}: p=0.5\) that the coin is balanced. (b) A local candidate contacts an SRS of 900 of the registered voters in his district to see if there is evidence that more than half support the bill he is sponsoring. (c) A college president says, " \(99 \%\) of the alumni support my firing of Coach Boggs." You contact an SRS of 200 of the college's 15,000 living alumni to test the hypothesis \(H_{0}: p=0.99\).
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