A report summarizes a survey of people in two independent random samples. One sample consisted of 800 young adults (age 19 to 35), and the other sample consisted of 400 parents of young adults age 19 to 35. The young adults were presented with a variety of situations (such as getting married or buying a house) and were asked if they thought that their parents were likely to provide financial support in that situation. The parents of young adults were presented with the same situations and asked if they would be likely to provide financial support to their child in that situation.
The report stated that the proportion of young adults who thought their parents would help with buying a house or renting an apartment for the sample of young adults was 0.37. For the sample of parents, the proportion who said they would help with buying a house or renting an apartment was 0.27. Based on these data, can you conclude that the proportion of parents who say they would help with buying a house or renting an apartment is significantly less than the proportion of young adults who think that their parents would help? Test the appropriate hypotheses using a significance level of 0.05. (Let p1 be the proportion of all parents of young adults who say they would help with buying a house or renting an apartment, and p2 be the proportion of young adults who think that their parents would help.)
Find the test statistic and P-value.
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To test whether the proportion of parents who say they would help with buying a house or renting an apartment is significantly less than the proportion of young adults who think that their parents would help, we can conduct a two-sample z-test for proportions.
The null hypothesis, H0, is that the proportion of parents who say they would help with buying a house or renting an apartment is equal to or greater than the proportion of young adults who think their parents would help (p1 ≥ p2).
The alternative hypothesis, Ha, is that the proportion of parents who say they would help with buying a house or renting an apartment is less than the proportion of young adults who think their parents would help (p1 < p2).
Given that the sample size for young adults is 800 and the proportion is 0.37, we can calculate the number of young adults who think their parents would help:
Number of young adults = Sample size * Proportion = 800 * 0.37 = 296.
Given that the sample size for parents is 400 and the proportion is 0.27, we can calculate the number of parents who say they would help:
Number of parents = Sample size * Proportion = 400 * 0.27 = 108.
To find the test statistic and P-value, we can use the formula for the test statistic:
z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2)),
where p = (p1*n1 + p2*n2) / (n1 + n2), n1 and n2 are the sample sizes, and p1 and p2 are the proportions.
Plugging in the values:
p = (0.37 * 800 + 0.27 * 400) / (800 + 400) = 0.33,
n1 = 800,
n2 = 400,
p1 = 0.37,
p2 = 0.27.
z = (0.37 - 0.27) / sqrt(0.33 * (1 - 0.33) * (1/800 + 1/400)),
z = 0.1 / sqrt(0.33 * 0.67 * (0.00125 + 0.0025)),
z = 0.1 / sqrt(0.033 * 0.00375),
z = 0.1 / sqrt(0.00012375),
z ≈ 6.124.
To find the P-value, we can use a z-table or calculator to find the cumulative probability of the test statistic z being less than or equal to 6.124. Using a calculator, the P-value is approximately 1.0 (since the calculated z-value is far beyond the range captured in typical z-tables).
Since the P-value is greater than the significance level (0.05), we fail to reject the null hypothesis. There is not enough evidence to conclude that the proportion of parents who say they would help with buying a house or renting an apartment is significantly less than the proportion of young adults who think their parents would help.
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