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Pulse rates A medical researcher measured the pulse rates (beats per minute) of a sample of randomly selected adults and found the following Student's \(t\) -based confidence interval: With \(95.00 \%\) Confidence, $$ 70.887604<\mu(\text { Pulse })<74.497011 $$ a. Explain carefully what the software output means. b. What's the margin of error for this interval? c. If the researcher had calculated a \(99 \%\) confidence interval, would the margin of error be larger or smaller? Explain.

Crawling Data collected by child development scientists produced this confidence interval for the average age (in weeks) at which babies begin to crawl: t-Interval for \(\mu\). \(30.65<\mu(\) age \()<32.89\) \((95.00 \%\) Confidence) a. Explain carefully what the software output means. b. What is the margin of error for this interval? c. If the researcher had calculated a \(90 \%\) confidence interval, would the margin of error be larger or smaller? Explain.

Cholesterol In the latest National Health and Nutrition Examination Survey (NHANES 2013/2014-wwwn.cdc.gov/ nchs/nhanes), HDL cholesterol of 2515 U.S. adults averaged \(53.9 \mathrm{mg} / \mathrm{dL}\) with a standard deviation of \(16.2729 \mathrm{mg} / \mathrm{dL}\). (Data in NHANES) a. Can you apply the Central Limit Theorem to describe the distribution of the cholesterol measurements? Why or why not? b. Can you apply the Central Limit Theorem to describe the sampling distribution model for the sample mean of U.S. adults? Why or why not? c. Sketch and clearly label the sampling model of the mean cholesterol levels of samples of size 2515 based on the \(68-95-99.7\) Rule.

Parking Hoping to lure more shoppers downtown, a city builds a new public parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period (44 weekdays), daily fees collected averaged \(\$ 126,\) with a standard deviation of \(\$ 15 .\) a. What assumptions must you make in order to use these statistics for inference? b. Write a \(90 \%\) confidence interval for the mean daily income this parking garage will generate. c. Interpret this confidence interval in context. d. Explain what "90\% confidence" means in this context. e. The consultant who advised the city on this project predicted that parking revenues would average \(\$ 130\) per day. Based on your confidence interval, do you think the consultant was correct? Why?

Speed of light In \(1882,\) Michelson measured the speed of light (usually denoted \(c\) as in Einstein's famous equation \(E=m c^{2}\) ). His values are in \(\mathrm{km} / \mathrm{sec}\) and have 299,000 subtracted from them. He reported the results of 23 trials with a mean of 756.22 and a standard deviation of 107.12 . a. Find a \(95 \%\) confidence interval for the true speed of light from these statistics. b. State in words what this interval means. Keep in mind that the speed of light is a physical constant that, as far as we know, has a value that is true throughout the universe. c. What assumptions must you make in order to use your method?

GPAs A college's data about the incoming freshmen indicate that the mean of their high school GPAs was \(3.4,\) with a standard deviation of 0.35 ; the distribution was roughly mound-shaped and only slightly skewed. The students are randomly assigned to freshman writing seminars in groups of 25\. What might the mean GPA of one of these seminar groups be? Describe the appropriate sampling distribution model-shape, center, and spread-with attention to assumptions and conditions. Make a sketch using the \(68-95\) \(99.7 \mathrm{P}\)

Pregnancy Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days. a. What percentage of pregnancies should last between 270 and 280 days? b. At least how many days should the longest \(25 \%\) of all pregnancies last? c. Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let \(\bar{y}\) represent the mean length of their pregnancies. According to the Central Limit Theorem, what's the distribution of this sample mean, \(\bar{y} ?\) Specify the model, mean, and standard deviation. d. What's the probability that the mean duration of these patients' pregnancies will be less than 260 days?

Rainfall Statistics from Cornell's Northeast Regional Climate Center indicate that Ithaca, New York, gets an average of 35.4 " of rain each year, with a standard deviation of \(4.2 " .\) Assume that a Normal model applies. a. During what percentage of years does Ithaca get more than \(40^{\prime \prime}\) of rain? b. Less than how much rain falls in the driest \(20 \%\) of all years? c. A Cornell University student is in Ithaca for 4 years. Let \(\bar{y}\) represent the mean amount of rain for those 4 years. Describe the sampling distribution model of this sample mean, \(\bar{y}\). d. What's the probability that those 4 years average less than \(30^{\prime \prime}\) of rain?

At work Some business analysts estimate that the length of time people work at a job has a mean of 6.2 years and a standard deviation of 4.5 years. a. Explain why you suspect this distribution may be skewed to the right. b. Explain why you could estimate the probability that 100 people selected at random had worked for their employers an average of 10 years or more, but you could not estimate the probability that an individual had done so.

Doritos Some students checked 6 bags of Doritos marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): \(29.2,28.5,28.7,28.9,29.1,29.5 .\) a) Do these data satisfy the assumptions for inference? Explain. b) Find the mean and standard deviation of the weights. c) Create a \(95 \%\) confidence interval for the mean weight of such bags of chips. d) Explain in context what your interval means. e) Comment on the company's stated net weight of 28.3 grams. \({ }^{\star}\) f) \(\quad\) Why might finding a bootstrap confidence interval not be a good idea for these data?