Question

According to NPR, in 2016 32.1% of adults aged 18-34 lived at home with their parents. a sociologist recently randomly surveyed 500 people aged 18-34 and found 143 of them did. at a=0.05, do we think the proportion has changed? Use the Null and alternative method Test Statistic Critical Region Decision about Null hypothesis conclusion

Answer 1

To test the proportion of adults aged 18-34 living at home with their parents, we can conduct a hypothesis test with the null and alternative hypotheses. Using the data provided, we calculate the test statistic (z-score) and compare it to the critical region at an alpha level of 0.05. Based on the results, we fail to reject the null hypothesis, indicating that there is not enough evidence to conclude that the proportion has changed.

Step-by-step explanation:

To test whether the proportion of adults aged 18-34 living at home with their parents has changed, we can conduct a hypothesis test. Let's define the null and alternative hypotheses:

H0: The proportion of adults aged 18-34 living at home with their parents is 32.1% (the same as in 2016)

HA: The proportion of adults aged 18-34 living at home with their parents has changed

To determine if there is enough evidence to reject the null hypothesis, we need to compute the test statistic and compare it to the critical region:

1. Compute the test statistic (z-score):

The test statistic is calculated using the formula:

z = (p - P) / sqrt((P(1-P)) / n)

Where:

p is the sample proportion (143/500 = 0.286)

P is the hypothesized proportion from H0 (0.321)

n is the sample size (500)

Substituting the values into the formula:

z = (0.286 - 0.321) / sqrt((0.321(1-0.321)) / 500)

z = -0.035 / sqrt(0.214 / 500)

z = -0.035 / sqrt(0.000428)

z ≈ -1.91

1. Find the critical region:

Since the alternative hypothesis is that the proportion has changed, we will perform a two-tailed test. With an alpha level of 0.05, the critical regions are at -1.96 and 1.96.

1. Decision about the Null hypothesis:

The test statistic (-1.91) is not in the critical region, which means we fail to reject the null hypothesis. Therefore, we do not have enough evidence to conclude that the proportion of adults aged 18-34 living at home with their parents has changed.

Conclusion:

Based on the hypothesis test, at a significance level of 0.05, we do not think the proportion of adults aged 18-34 living at home with their parents has changed.