Problem 1: Adults saving for retirement
In a recent survey conducted by Pew Research, it was found that 156 out of 295 adult Americans without a high school diploma were worried about having enough saved for retirement. Does the sample evidence suggest that a majority of adult Americans without a high school diploma are worried about having enough saved for retirement? Use a 0.05 level of significance.
State the null and alternative hypothesis:
Null hypothesis (H0): The proportion of adult Americans without a high school diploma worried about having enough saved for retirement is equal to 0.5.
Alternative hypothesis (Ha): The proportion of adult Americans without a high school diploma worried about having enough saved for retirement is greater than 0.5.
What type of hypothesis test is to be used?
A one-sample proportion hypothesis test is to be used.
What distribution should be used and why?
The normal distribution should be used because the sample size is large enough (295) and the data is assumed to be independent.
Is this a right, left, or two-tailed test?
This is a right-tailed test because we are testing if the proportion is greater than 0.5.
Compute the test statistic:
The test statistic can be calculated using the formula:
z = (p̂ - p0) / √(p0(1-p0)/n)
where p̂ is the sample proportion, p0 is the hypothesized proportion, and n is the sample size.
Compute the p-value:
The p-value can be calculated by finding the area under the normal distribution curve to the right of the test statistic.
Do you reject or not reject the null hypothesis? Explain why.
If the p-value is less than the significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
What do you conclude?
Based on the sample evidence, we can conclude that there is sufficient evidence to suggest that a majority of adult Americans without a high school diploma are worried about having enough saved for retirement.