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 p be the proportion of
all parents of young adults who say they would help with buying a house or renting an apartment, and p
be the proportion of young adults who think that their parents would help.)
a) State the appropriate null and alternative hypotheses.
b) Find the test statistic and P-value. (
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a) The appropriate null and alternative hypotheses are:
- Null hypothesis (H0): The proportion of parents who say they would help with buying a house or renting an apartment is equal to the proportion of young adults who think their parents would help.
- Alternative hypothesis (H1): 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.
b) To find the test statistic and p-value, we can use the two-sample z-test for comparing proportions.
First, we need to calculate the standard errors of the sample proportions:
For the sample of young adults:
n1 = 800 (sample size of young adults)
p1 = 0.37 (proportion of young adults who think their parents would help)
std_error1 = sqrt(p1*(1-p1)/n1)
For the sample of parents:
n2 = 400 (sample size of parents)
p2 = 0.27 (proportion of parents who say they would help)
std_error2 = sqrt(p2*(1-p2)/n2)
Next, we can calculate the test statistic (z-score) using the formula:
test_stat = (p1 - p2) / sqrt(std_error1^2 + std_error2^2)
Finally, we can calculate the p-value corresponding to the test statistic, assuming a normal distribution:
p-value = P(Z < test_stat)
Using the provided data and formulas, the test statistic and p-value can be calculated.
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