Question

question 9
a sociologist conducts a hypothesis test because she has a theory that the proportion of college students who study abroad falls below 0.45. after surveying a random sample of college students, and conducting a hypothesis test, the sociologist reports a p - value of 0.01. what can we conclude based on this information?
the sample proportion is at least three standard deviations away from 0.45.
there must be a large difference between p and \\( \hat { p } \\).
if \\( \alpha \\) is set at 0.01, the researcher should conclude there is no evidence against the null hypothesis.
the sample size must have been large.
none of the above answer options are correct

Explanation:

• For a hypothesis test, if the \(P -\)value is less than or equal to the significance level \(\alpha\), we reject the null hypothesis. Here, when \(\alpha = 0.01\) and \(P=0.01\), we reject the null hypothesis (not conclude there is no evidence against it).
• The \(P -\)value does not directly tell us about the number of standard deviations (a \(P -\)value of \(0.01\) in a one - tailed test corresponds to a \(z -\)score of approximately \(z=- 2.33\) (not at least \(3\) standard deviations).
• The \(P -\)value is a function of the sample proportion \(\hat{p}\), sample size \(n\), and the hypothesized proportion \(p\). A small \(P -\)value does not necessarily mean a large difference between \(p\) and \(\hat{p}\) (it could also be due to a large sample size). And a small \(P -\)value does not guarantee a large sample size (it depends on the combination of \(\hat{p}-p\) and \(n\)).

Answer:

None of the above answer options are correct.