Question

1. a publication states that 80% of college students own a laptop, but a researcher suspects this value is not high enough. when a random sample of 250 students is surveyed, 84% of the sample reports owning a laptop. based on this information, which one of the following statements is correct?

a. there is evidence against the null hypothesis since 0.84 is larger than 0.80.
b. if the researcher computes a negative value of z, this means an error was made when calculating the test statistic.
c. as long as the p - value is large, the researcher should conclude that there is evidence against the null hypothesis.
d. the hypothesis test will determine whether it is true that p = 0.84.
e. none of the above statements are correct.

Explanation:

Step1: Analyze Option A

Just because the sample proportion \(0.84\) is larger than the hypothesized proportion \(0.80\) does not necessarily mean there is evidence against the null hypothesis. We need to consider the sampling variability. A hypothesis - test (using a z - test for proportions: \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(\hat{p}=0.84\), \(p = 0.80\), \(n = 250\)) is required to determine if the difference is statistically significant.

Step2: Analyze Option B

The formula for the z - statistic in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\). If \(\hat{p}

Step3: Analyze Option C

If the \(P\) - value is large, we fail to reject the null hypothesis. The rule is: if \(P\) - value \(<\alpha\) (significance level, usually \(\alpha=0.05\)), we reject the null hypothesis. A large \(P\) - value does not provide evidence against the null hypothesis.

Step4: Analyze Option D

The hypothesis test is to test the null hypothesis \(H_0:p = 0.80\) against an alternative hypothesis (in this case, \(H_1:p>0.80\)). It is not to test if \(p = 0.84\). The sample proportion \(\hat{p}=0.84\) is just an estimate of the population proportion \(p\).

Answer:

E. None of the above statements are correct.