91Ó°ÊÓ

A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A \(95 \%\) confidence interval based on this sample is (128 minutes, 147 minutes), which is based on the normal model for the mean. Determine whether the following statements are true or false, and explain your reasoning. (a) We are \(95 \%\) confident that the average waiting time of these 64 emergency room patients is between 128 and 147 minutes. (b) We are \(95 \%\) confident that the average waiting time of all patients at this hospital's emergency room is between 128 and 147 minutes. (c) \(95 \%\) of random samples have a sample mean between 128 and 147 minutes. (d) A \(99 \%\) confidence interval would be narrower than the \(95 \%\) confidence interval since we need to be more sure of our estimate. (e) The margin of error is 9.5 and the sample mean is 137.5 . (f) In order to decrease the margin of error of a \(95 \%\) confidence interval to half of what it is now, we would need to double the sample size. (Hint: the margin of error for a mean scales in the same way with sample size as the margin of error for a proportion.)

Write the null and alternative hypotheses in words and using symbols for each of the following situations. (a) Since 2008 , chain restaurants in California have been required to display calorie counts of each menu item. Prior to menus displaying calorie counts, the average calorie intake of diners at a restaurant was 1100 calories. After calorie counts started to be displayed on menus, a nutritionist collected data on the number of calories consumed at this restaurant from a random sample of diners. Do these data provide convincing evidence of a difference in the average calorie intake of a diners at this restaurant? (b) The state of Wisconsin would like to understand the fraction of its adult residents that consumed alcohol

400 students were randomly sampled from a large university, and 289 said they did not get enough sleep. Conduct a hypothesis test to check whether this represents a statistically significant difference from \(50 \%\), and use a significance level of 0.01 .

In each part below, there is a value of interest and two scenarios (I and II). For each part, report if the value of interest is larger under scenario I, scenario II, or whether the value is equal under the scenarios. (a) The standard error of \(\hat{p}\) when (I) \(n=125\) or (II) \(n=500\). (b) The margin of error of a confidence interval when the confidence level is (I) \(90 \%\) or (II) \(80 \%\). (c) The p-value for a Z-statistic of 2.5 calculated based on a (I) sample with \(n=500\) or based on a (II) sample with \(n=1000\). (d) The probability of making a Type 2 Error when the alternative hypothesis is true and the significance level is (I) 0.05 or (II) 0.10 .