You work for the public relations department of the Social Security Administration. In an effort to design better advertising campaigns, your department decides to conduct a survey to find out the opinions people in the United States have about the Social Security system. One of the questions asked and the respondent’s age are shown in the table below.
Question: Do you think the Social Security system will have the money available to provide the benefits you expect for your retirement?
Results:
AGE RESPONSE AGE RESPONSE AGE RESPONSE AGE RESPONSE AGE RESPONSE
83 Yes 46 No 50 Yes 35 Yes 30 Yes
37 No 25 No 76 Yes 47 No 71 No
49 No 30 No 51 Yes 29 Yes 64 No
27 Yes 72 Yes 58 No 76 No 18 No
27 No 66 No 79 Yes 31 Yes
59 Yes 47 No 69 Yes 40 No
44 Yes 47 No 66 No 61 Yes
1. What sampling technique would you use to select the sample for the study? Why? What sampling technique would you use if you wanted to select samples from four age groups: 18-29, 30-49, 50-64, and 65+ ?
2. What technique in part (a) will give you a sample that is representative of the population?
3. Identify possible flaws or biases in your study.
4. Test the claim that less than 40% of people in the United States think the Social Security system will have the money available to provide the benefits they expect for their retirements. Use á = 0.10. Write a paragraph that interprets the test’s decision; Does the decision support your department’s claim?
5. Test the claim that the mean age of people in the United States who would say yes to the survey question shown above is 60 years or older. Use á = 0.10 and assume that the population is normally distributed. Write a paragraph that interprets the test’s decision. Is there enough evidence to reject your department’s claim?
6. Your conclusions – on the basis of your analysis of the responses to this survey question, what would you tell your department?
We do not do your work for you. Once you have answered your questions, we will be happy to give you feedback on your work. Although it might require more time and effort, you will learn more if you do your own work. Isn't that why you go to school?
1. The sampling technique used in this study is not mentioned in the given information. However, a simple random sampling technique would typically be used to select the sample for this kind of study. This ensures that every individual in the population has an equal chance of being selected.
If the goal is to select samples from four age groups (18-29, 30-49, 50-64, and 65+), then a stratified sampling technique can be used. In stratified sampling, the population is divided into mutually exclusive groups (strata), and samples are selected from each stratum in proportion to their representation in the population. In this case, individuals from each age group can be randomly selected to form the samples.
2. Stratified sampling is a technique that will give you a sample that is representative of the population. By dividing the population into age groups and selecting samples from each group, the resulting sample will have individuals from different age groups in proportion to their representation in the population. This helps ensure that the sample is more representative and reduces the risk of biases.
3. Possible flaws or biases in this study could include:
- Non-response bias: If not all individuals who were asked to participate in the survey responded, the opinions of those who did respond may not be representative of the entire population.
- Self-selection bias: Individuals who chose to participate in the survey may have different opinions compared to those who chose not to participate, leading to a biased sample.
- Age bias: The sample may not accurately represent the distribution of ages in the population, leading to biased results.
4. To test the claim that less than 40% of people in the United States think the Social Security system will have the money available to provide the benefits they expect for retirement, a hypothesis test can be conducted. The null hypothesis (H0) would be that the proportion is equal to or greater than 40%, while the alternative hypothesis (Ha) would be that the proportion is less than 40%. A one-sample proportion z-test can be utilized, with a significance level (α) of 0.10.
By calculating the test statistic and comparing it to the critical value or the p-value, a decision can be made. If the test statistic falls in the rejection region (rejecting the null hypothesis), then the decision would support the claim that less than 40% of people in the United States believe in the Social Security system's financial availability for retirement benefits.
5. To test the claim that the mean age of people who would respond "yes" to the survey question is 60 years or older, a one-sample t-test can be used assuming the population is normally distributed. The null hypothesis (H0) would be that the mean age is 60 or younger, while the alternative hypothesis (Ha) would be that the mean age is older than 60. The significance level (α) is set at 0.10.
After calculating the test statistic and comparing it to the critical value or the p-value, a decision can be made. If the test statistic falls in the rejection region (rejecting the null hypothesis), there is enough evidence to reject the department's claim that the mean age of respondents is 60 years or older.
6. Based on the analysis of the responses to this survey question, I would inform the department that there is a mix of opinions regarding the Social Security system's financial availability for retirement benefits. The survey results show that individuals of various ages have different beliefs about the system's future ability to provide expected benefits. It is important for the department to consider these differences in opinions when designing advertising campaigns to effectively communicate and address concerns related to the Social Security system. Additionally, the department should be aware of possible biases and limitations in the study, such as non-response and self-selection biases, which may affect the generalizability of the findings.