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 use a hypothesis test. The null hypothesis (H0) is that there has been no change in the percentage of drivers who enjoy driving their cars, and the alternative hypothesis (Ha) is that the percentage has decreased.

H0: The percentage of drivers who enjoy driving their cars is equal to 79%
Ha: The percentage of drivers who enjoy driving their cars is less than 79%

To calculate the large sample z statistic, we need the sample proportion and the standard error.
Sample proportion (p̂) = 69% = 0.69
Sample size (n) = 1048

To calculate the standard error, we need to use the formula:

Standard error (SE) = √[(p̂(1-p̂))/n]

SE = √[(0.69(1-0.69))/1048]
SE ≈ 0.014

Next, we can calculate the z statistic using the formula:

z = (p̂ - p) / SE

Where p is the population proportion under the null hypothesis. In this case, p = 0.79.

z = (0.69 - 0.79) / 0.014
z ≈ -7.14

The z statistic is approximately -7.14.

To find the p-value associated with this z statistic, we can use a standard normal distribution table or calculator. Since the alternative hypothesis is that the percentage has decreased, we want to find the probability of getting a z statistic less than -7.14 (left-tailed test).

The p-value is the probability of obtaining a z score of -7.14 or less in a standard normal distribution. This probability represents the likelihood of observing a sample proportion as extreme as or more extreme than 0.69, assuming the null hypothesis is true.

By looking up the z statistic in a standard normal distribution table or using a z-table calculator, we can find that the p-value is approximately 0.

Therefore, the large sample z statistic is approximately -7.14, and the p-value is approximately 0.

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