Answer:
Explanation:
a. Variables needed to solve the problem:
Sample size: n = 1,026
Proportion of the sample that responded "Make some major changes": p = 0.39
Confidence level: 95%
b. To determine if a normal distribution can be used to approximate the data, we need to check if the sample size is large enough to meet the requirements for a normal approximation. The sample size should be at least 10 times larger than the number of successes (np) and 10 times larger than the number of failures (n(1-p)). In this case, we have:
np = 1026 x 0.39 = 399.14
n(1-p) = 1026 x 0.61 = 626.86
Both np and n(1-p) are greater than 10, so we can assume that a normal distribution can be used to approximate the data.
c. The standard deviation of the proportion can be calculated using the following formula:
standard deviation = sqrt(p(1-p) / n)
standard deviation = sqrt(0.39 x 0.61 / 1026) = 0.024
d. The point estimate of the proportion of all United States adults who would respond "Make some major changes" is simply the sample proportion, which is p = 0.39. The margin of error can be calculated using the following formula:
margin of error = z* * standard deviation
where z* is the z-score associated with the 95% confidence level. Using a standard normal distribution table or a calculator, we find that the z-score for a 95% confidence level is approximately 1.96. Therefore:
margin of error = 1.96 * 0.024 = 0.047
e. The confidence interval can be calculated using the following formula:
confidence interval = point estimate ± margin of error
confidence interval = 0.39 ± 0.047
confidence interval = (0.343, 0.437)
Therefore, we are 95% confident that the proportion of all United States adults who would respond "Make some major changes" is between 0.343 and 0.437.