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Question: a study aims to understand the yeartoyear changes in small...

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A study aims to understand the year-to-year changes in small business sales during the winter holiday period in the US. Researchers randomly sampled 100 small businesses in December 2014 and found that 66 of them had an increase in sales relative to December 2013. a. Find a 95% confidence interval for the proportion of businesses that had an increase in sales. Find the sample size needed for the 68% confidence interval to be of the same width as the 95% confidence interval calculated above (round up the sample size to the nearest integer if necessary). Note that the 95% confidence interval calculated above was based on a sample of size 100. Another research paper (funded by an agency known to be critical of the governments policies) cited this study claiming that less than 2/3rd of the small businesses had an increase in sales. With what level of confidence can this statement be made? [10 pt] Similar to the study done for December 2014, if a 95% confidence interval was computed for each of the months starting from January 2011 to December 2014 (that is 4x12-48 months), what is the probability that at least 95% of the confidence intervals cover the true proportions? Feel free to assume (and state clearly), if required, that the sales for each small business for each month are independent of that for any other combination of small business and month. Clearly specify if, where, and how you have used this assumption. Hint: As per our definition in Lecture 6, a 95% confidence interval covers the true proportion 95% of the times. b. c. d.

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