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Parks and Recreation A questionnaire was mailed to 1000 registered municipal voters selected at random. Only 500 questionnaires were returned, and of the 500 returned, 360 respondents were strongly opposed to a surcharge proposed to support the city Parks and Recreation Department. Are you willing to accept the \(72 \%\) figure as a valid estimate of the percentage in the city who are opposed to the surcharge? Why or why not?

Short Answer

Expert verified
Answer: Considering the 50% non-response rate and the potential bias it could introduce, we should be cautious about accepting the 72% figure as a valid estimate. It is important to have a more significant response rate or conduct further studies to have better confidence in the results.

Step by step solution

01

Analyze the survey data

From the information provided, we can see that of 1000 voters, only 500 questionnaires were returned. Out of these 500 returned, 360 were strongly opposed to the surcharge for the Parks and Recreation Department. We can calculate the percentage of returned questionnaires that are opposed to the surcharge with the following formula: Percentage = (Number of Opposed Respondents / Total Number of Returned Questionnaires) × 100
02

Calculate the percentage opposed

Using the data provided, we can now calculate the percentage of returned questionnaires that were opposed to the surcharge: Percentage = (360 / 500) × 100 = 72% So the percentage of opposed respondents among the returned questionnaires is 72%.
03

Discuss if the sample is representative

Now we need to discuss if this 72% is a valid estimate of the percentage of the entire city population who are opposed to the surcharge. Since only 500 out of the 1000 questionnaires were returned, this means we have a 50% response rate, meaning that we have only information from half of the randomly chosen voters. This could introduce bias in the survey results, as non-respondents might hold different opinions than respondents. Furthermore, the total number of municipal voters might be much larger than the 1000 sampled, leading to some questions about the representativeness of the survey. Although it's typical for surveys to have non-responses, 50% is quite a high rate of non-responses, which may weaken the conclusion if these non-respondents have different opinions.
04

Give the final conclusion

Considering the 50% non-response rate and the potential bias it could introduce, we should be cautious about accepting the 72% figure as a valid estimate of the percentage of the entire city who are opposed to the surcharge. It is important to have a more significant response rate or conduct further studies to have better confidence in the results.

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Key Concepts

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

Response Rate
The "Response Rate" in a survey context refers to the proportion of individuals who respond to a survey or questionnaire out of the total number asked. In this exercise, the response rate can be calculated as the number of returned questionnaires divided by the number of people who received a questionnaire.

For this survey, 500 out of 1000 potential respondents returned their questionnaires. This indicates a response rate of 50%. A response rate is important because it affects the reliability of the survey results. A higher response rate often reflects more reliable results since it suggests that a larger segment of the audience has been reached and their opinions considered.

Reasons why the response rate is crucial:
  • Reflects how engaged or interested the population is in the subject matter.
  • Higher rates decrease the potential for non-response bias, where the opinions of those who didn't respond could differ substantively from those who did.
  • Aids in understanding the representativeness of the sample.
In some cases, a 50% response rate may still be adequate, but it’s essential to consider potential biases brought by non-responses.
Sampling Bias
Sampling Bias arises when the method of selecting a sample causes certain outcomes to be overrepresented and others to be underrepresented. This can skew results and lead to misleading conclusions. In this exercise, the concern is whether the 72% opposition to the surcharge accurately represents the views of all municipal voters.

Possible causes of sampling bias here include:
  • Non-response bias: Since only 50% of voters responded, the opinions of the other 50% remain unknown. Those who chose not to respond might have different views.
  • Voluntary response bias: Those who feel strongly (either in favor or against the surcharge) might be more motivated to respond, potentially skewing results.

Researchers must evaluate whether the returned sample faithfully represents the entire voter base. Mitigating sampling bias is essential to ensure valid and generalizable survey results.
Statistical Validity
Statistical Validity ensures that the results of a survey or experiment truly reflect what is happening in the wider population. Validity considers various factors including response rate, sampling methods, and potential biases.

In the context of the given survey, the 72% figure, derived from the respondents, needs to be assessed for validity.
  • The 50% response rate suggests caution. If non-respondents have different views, the extracted data may lack validity.
  • Sampling bias might distort conclusions. This can happen if the sample does not accurately reflect the overall population.

To enhance statistical validity, additional measures might include follow-up surveys, larger sample sizes, or validation against other data sources. Reliable conclusions stem from ensuring that the sample and methods capture the reality of the larger group accurately.

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Most popular questions from this chapter

Parents with children list a GPS system \((28 \%)\) and a DVD player \((28 \%)\) as "must have" accessories for a road trip. \({ }^{12}\) Suppose a sample of \(n=1000\) parents are randomly selected and asked what devices they would like to have for a family road trip. Let \(\hat{p}\) be the proportion of parents in the sample who choose either a GPS system or a DVD player. a. If \(p=.28+.28=.56,\) what is the exact distribution of \(\hat{p}\) ? How can you approximate the distribution of \(\hat{p} ?\) b. What is the probability that \(\hat{p}\) exceeds. \(6 ?\) c. What is the probability that \(\hat{p}\) lies between. .5 and \(6 ?\) d. Would a sample percentage of \(\hat{p}=.7\) contradict the reported value of \(.56 ?\)

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