Question

According to a large poll in a previous year, about 80% of homes in a certain county had access to high-speed internet. Market researchers wondered if that proportion had changed, so they took a random sample of 64 homes from that county and found that 48 of them had access to high-speed internet. They want to use this sample data to test H0:p=0.8 versus Ha:p≠0.8, where p is the proportion of homes in this county with high-speed internet access. Assuming that the conditions for inference have been met, calculate the test statistic for their significance test. You may round to two decimal places.

Answer 1

The test statistic for the significance test to determine if the proportion of homes with high-speed internet access in the county has changed from 80% is calculated using the 1-Proportion Z-test, which yields a test statistic (Z) of -1.00.

Step-by-step explanation:

The market researchers are using hypothesis testing to determine if the proportion of homes with high-speed internet access in the county has changed from the previously reported 80%. With a random sample of 64 homes where 48 have high-speed internet, we can perform a hypothesis test using the 1-Proportion Z-test.

To calculate the test statistic, we use the formula:
Z = (p-hat - p0) / sqrt(p0(1-p0)/n)
where p-hat is the sample proportion, p0 is the hypothesized population proportion, and n is the sample size.

In this case, p-hat is 48/64 = 0.75, p0 is 0.8, and n is 64.
The test statistic (Z) is therefore calculated as:
Z = (0.75 - 0.8) / sqrt(0.8*0.2/64) = -1.00
The test statistic of -1.00 can now be used to determine the p-value or to compare against critical values of the Z-distribution at the desired significance level to reach a conclusion in the hypothesis test.