When calculating confidence intervals, you may use your calculator's interval features. Show what you typed in your calculator to get your answer. You are not expected to use formulas. When interpreting intervals, remember your "script." Include the confidence level, the parameter you are estimating, and numerical values with appropriate units. 1. A researcher wishes to determine the mean cost of a monthly heating bill for an American family. In a sample of 60 families, the mean was $165 and the standard deviation was $35. a. Find the 95% confidence interval for the mean cost of a heating bill for an American family. Round to the nearest hundredth. b. Interpret your result in a sentence. c. Assume that you had to find a 95% confidence interval for the same data, but with sample size 90. Assume the sample mean and population standard deviation are the same as proposed in the problem above. What is the effect on the width of the confidence interval? d. Assume we are given the same sample mean, population standard deviation, and confidence level. Without calculating, determine which sample size would produce the widest interval: 30, 60, or 90? Explain why this occurs.2. The 99% confidence interval for the average number of hours of sleep the average American gets each night is (6.08,7.52). You should be able to do these problems only with arithmetic; no statistics formulas are needed. Show work for your calculations. a. Find the error bound (margin of error). See this video hint for help. b. Find the sample mean. 3. A researcher wishes to estimate the mean GPA in a certain college with 95% confidence. The GPA of a sample of 32 students is shown. 2.31 3.30 3.12 2.61 1.83 2.65 3.18 3.96 3.84 2.98 3.46 3.09 2.67 4.00 1.73 2.21 4.00 3.68 3.81 2.93 2.84 3.42 2.05 3.07 3.67 2.33 2.27 3.15 1.98 3.68 4.00 3.35 a. Find the 99% confidence interval for the mean GPA at this college. Round to the nearest hundredth. b. Interpret your interval in a sentence. c. An administrator has data from this population of students from 15 years ago showing that the mean GPA was 3.29. Is there sufficient evidence to believe that this mean GPA has changed? Explain.4. A sample of 25 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.6 pounds with a standard deviation of 0.32 pounds. The population standard deviation is known to be 0.49 pounds. a. Construct a 90% confidence interval for the population mean weight of the heads of lettuce. Round to the nearest hundredth. Interpret this interval in a sentence. b. Construct a 95% confidence interval for the population mean weight of the heads of lettuce. Explain the effects that you notice in increasing the confidence level from 90% to 95%. c. Imagine you constructed a 90% confidence interval, 95% confidence interval, and 98% confidence interval (with the same sample size, sample mean, and population standard deviation). Without calculating, determine which would be the widest interval? Explain why. d. Your classmate from statistics class was absent today and is asking for your help. He comments, "We already determined that the mean weight is 2.6 pounds. What's the point of constructing a confidence interval? Didn't we already find the mean we wanted?" Answer your classmate's question.5. A random sample of 501 drivers were asked, "Do you use a hands-free device when talking on your cellular phone in the car?" The survey found that 472 drivers did use a hands-free device. a. Estimate, with 98% confidence, the proportion of drivers who use a hands-free device when talking on a cellular phone in their car. Round decimals to the nearest thousandth and percents to the nearest tenth of a percent (example: 0.345 or 34.5%). b. Write a sentence to interpret this confidence interval. c. What is the margin of error in this confidence interval? See this video hint for help.6. The Gallup organization conducted a poll in October 2021 asking 823 American adults for their opinions on gun legislation. One question asked respondents if they feel the laws governing the sale of firearms should be made more strict, less strict, or kept as they are now. In this survey, 52% of those surveyed said laws should be more strict. a. Estimate, with 95% confidence, the percent of Americans who believe firearm legislation should be more strict. Round decimals to the nearest thousandth and percents to the nearest tenth of a percent (example: 0.345 or 34.5%). b. Interpret this interval in a sentence. c. A poll asking the same question was conducted in 2004, and the proportion of those suveyed who felt firearm legislation should be more strict was 60%. Is there sufficient evidence to conclude that the opinions of Americans on this topic changed between 2004 and 2021? Explain