In A Recent Poll Conducted By The Gallup Organization From
The 99% confidence interval is a statistical tool used to estimate the range within which a population parameter is likely to fall. The z-score at this confidence level is 2.807. The value of the z-score is determined by the chosen confidence level, such as 1.28 for an 80% confidence level and 1.96 for a 95% confidence level. Z-scores are equivalent to confidence levels, meaning that a z-score of 1.96 indicates a 95% confidence level. For a 99% confidence interval, the z-value is 1.96. This value can also be calculated for other confidence levels, such as 1.28 for an 80% confidence level.
a. To find the 99% confidence interval for the population proportion of Americans who believe in the lasting impact of the #MeToo movement, we can use the formula:
[ \text{Confidence Interval} = \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} ]
Where:
- (\hat{p}) is the sample proportion
- (Z) is the Z-score for a 99% confidence level, which is approximately 2.807
- (n) is the sample size
Let's calculate the confidence interval.
b. The verbal interpretation of the confidence interval would be: "We are 99% confident that the true proportion of Americans who believe in the lasting impact of the #MeToo movement lies between [lower bound] and [upper bound]."
c. The error for the confidence interval is the margin of error, which is calculated as follows:
[ \text{Margin of Error} = Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} ]
d. Based on the confidence interval, we can compare it with Walmart's statement. If the 60% falls within the confidence interval, we can say that the survey results are consistent with the Gallup poll. If it doesn't, we could advise Walmart to reconsider their survey methodology or sample representativeness.
Let's proceed with the calculations to find the confidence interval.
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