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 determine if there is evidence to support the claim that the percent of drivers who enjoy driving their cars has declined since 1991, we can conduct a hypothesis test using the information provided.
Let's define our hypotheses:
Null hypothesis (H0): The percent of drivers who enjoy driving their cars has not declined since 1991.
Alternative hypothesis (Ha): The percent of drivers who enjoy driving their cars has declined since 1991.
We can use the large sample z-test to compare the sample proportion (p) from the recent poll to the hypothesized proportion (p0) from 1991.
The formula for the z-test statistic is:
z = (p - p0) / sqrt(p0 * (1 - p0) / n)
Where:
p = sample proportion from the recent poll
p0 = hypothesized proportion from 1991
n = sample size from the recent poll
Let's calculate the z-test statistic:
Given:
p = 0.69
p0 = 0.79
n = 1048
z = (0.69 - 0.79) / sqrt(0.79 * (1 - 0.79) / 1048)
z ≈ -9.284
To find the p-value associated with this z-test statistic, we need to consult a standard normal distribution table or use a statistical calculator. The p-value is the probability of observing a z-value as extreme as -9.284 or more extreme in a standard normal distribution.
The p-value for z = -9.284 is extremely small (close to 0). This suggests strong evidence against the null hypothesis and in favor of the alternative hypothesis. Therefore, we can conclude that the percent of drivers who enjoy driving their cars has declined since 1991.
In summary, the large sample z statistic is approximately -9.284, and the associated p-value is very close to 0.