Part #1
Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in a national park. Then x has a distribution that is approximately normal with mean μ = 61.0 kg and standard deviation σ = 8.0 kg. Suppose a doe that weighs less than 52 kg is considered undernourished.
(a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (Round your answer to four decimal places.)
(b) If the park has about 2400 does, what number do you expect to be undernourished in December? (Round your answer to the nearest whole number.)
(c) To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 40 does should be more than 58 kg. If the average weight is less than 58 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight x for a random sample of 40 does is less than 58 kg (assuming a healthy population)? (Round your answer to four decimal places.)
(d) Compute the probability that x < 63 kg for 40 does (assume a healthy population). (Round your answer to four decimal places.)
Part #2
What price do farmers get for their watermelon crops? In the third week of July, a random sample of 41 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $2.00 per 100 pounds.
(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (Round your answers to two decimal places.)
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(b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.31 for the mean price per 100 pounds of watermelon. (Round up to the nearest whole number.)
(c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds. (Round your answers to two decimal places.)
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Part #3
Thirty-three small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 40.7 cases per year.
(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)
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(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)
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(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)
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