The pew research center recently polled n=1048 u.s. drivers and found that 69% enjoyed driving their cars.
In 1991 a poll reported this percent to be 79%. Using the data from this poll ,test the claim that the percent of drivers who enjoy driving their cars has declined since 1991. Report the large sample z statistic and its P value.
To test the claim that the percent of drivers who enjoy driving their cars has declined since 1991, we can conduct a hypothesis test using the large sample z-test.
Let's set up the null and alternative hypotheses:
Null Hypothesis (H0): The percent of drivers who enjoy driving their cars has not declined since 1991.
Alternative Hypothesis (H1): The percent of drivers who enjoy driving their cars has declined since 1991.
To obtain the large sample z statistic and its p-value, follow these steps:
Step 1: Calculate the sample proportion p̂ from the recent poll:
p̂ = (number of drivers who enjoy driving their cars) / (total number of drivers polled)
= 69% = 0.69 (expressed as a decimal)
Step 2: Calculate the standard error (SE) of the sample proportion:
SE = sqrt((p̂ * (1 - p̂)) / n)
= sqrt((0.69 * (1 - 0.69)) / 1048)
≈ 0.0153
Step 3: Calculate the z statistic using the formula:
z = (p̂ - p0) / SE, where p0 represents the claimed proportion (79%) from 1991.
z = (0.69 - 0.79) / 0.0153
≈ -6.54
Step 4: Calculate the p-value associated with the obtained z statistic:
For a two-tailed test, we need to find the probability of observing a z statistic that is as extreme or more extreme than the obtained value of -6.54.
Using a z-table or statistical software, we find that the p-value is very close to 0 (much smaller than the conventional significance level of 0.05).
Therefore, we can conclude that the p-value associated with the obtained z statistic is essentially zero.
To summarize:
The large sample z statistic is approximately -6.54, and its p-value is extremely close to 0. This suggests strong evidence to reject the null hypothesis. Hence, we can conclude that the percent of drivers who enjoy driving their cars has declined since 1991.