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Answer :
Final Answer:
1) Yes, at the 0.02 significance level, we can reject USA TODAY's claim.
2) The p-value is 0.00006, which indicates strong evidence against the claim that only 5% of U.S. adults have seen an extraterrestrial being.
3) The only type of error that could have occurred here is a Type I error.
4) The 98% confidence interval is (0.056, 0.124).
5) Yes, a proportion of 0.09 would be surprising given the sample proportion of 0.08.
Explanation:
1) To test whether USA TODAY's claim is accurate, we perform a hypothesis test using the sample proportion of those who have seen an extraterrestrial being. The resulting p-value is 0.00006, which is less than 0.02, the significance level we chose for our test. Therefore, we can reject USA TODAY's claim and conclude that the percentage of U.S. adults who have seen an extraterrestrial being is likely higher than 5%.
2) The p-value represents the probability of observing a sample proportion as extreme or more extreme than the one we obtained, given that the null hypothesis (USA TODAY's claim) is true. In this case, the small p-value indicates strong evidence against the null hypothesis and supports the conclusion that USA TODAY's claim is inaccurate.
3) A Type I error occurs when we reject a true null hypothesis. In this case, if the true percentage of U.S. adults who have seen an extraterrestrial being was actually 5%, but we rejected this null hypothesis based on our sample data, we would have committed a Type I error.
4) The 98% confidence interval for the population proportion is calculated as the sample proportion + or - the margin of error. The margin of error is found using the formula z*√((p-hat(1-p))/n), where z* is the z-value associated with a 98% confidence level. The resulting interval is (0.056, 0.124), which indicates that we are 98% confident that the true population proportion falls within this range.
5) Given that the sample proportion in our original sample was 0.08, a proportion of 0.09 in a new sample of 250 would not be surprising because it falls within the 98% confidence interval. This indicates that the sample proportion we obtained is not an anomaly and is likely representative of the true population proportion.
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