Problem 39
Based on the model \(N(1152,84)\) describing Angus steer weights, what are the cutoff values for a) the highest \(10 \%\) of the weights? b) the lowest \(20 \%\) of the weights? c) the middle \(40 \%\) of the weights?
Problem 40
In the Normal model \(N(100,16)\), what cutoff value bounds a) the highest \(5 \%\) of all IQs? b) the lowest \(30 \%\) of the IQs? c) the middle \(80 \%\) of the IQs?
Problem 41
Consider the Angus weights model \(N(1152,84)\) one last time. a) What weight represents the 40 th percentile? b) What weight represents the 99 th percentile? c) What's the IQR of the weights of these Angus steers?
Problem 42
Consider the IQ model \(N(100,16)\) one last time. a) What IQ represents the 15 th percentile? b) What IQ represents the 98 th percentile? c) What's the IQR of the IQs?
Problem 43
Assume the cholesterol levels of adult American women can be described by a Normal model with a mean of \(188 \mathrm{mg} / \mathrm{dL}\) and a standard deviation of \(24 .\) a) Draw and label the Normal model. b) What percent of adult women do you expect to have cholesterol levels over \(200 \mathrm{mg} / \mathrm{dL}\) ? c) What percent of adult women do you expect to have cholesterol levels between 150 and \(170 \mathrm{mg} / \mathrm{dL}\) ? d) Estimate the IQR of the cholesterol levels. e) Above what value are the highest \(15 \%\) of women's cholesterol levels?
Problem 44
A tire manufacturer believes that the treadlife of its snow tires can be described by a Normal model with a mean of 32,000 miles and standard deviation of 2500 miles. a) If you buy a set of these tires, would it be reasonable for you to hope they'll last 40,000 miles? Explain. b) Approximately what fraction of these tires can be expected to last less than 30,000 miles? c) Approximately what fraction of these tires can be expected to last between 30,000 and 35,000 miles? d) Estimate the IQR of the treadlives. e) In planning a marketing strategy, a local tire dealer wants to offer a refund to any customer whose tires fail to last a certain number of miles. However, the dealer does not want to take too big a risk. If the dealer is willing to give refunds to no more than 1 of every 25 customers, for what mileage can he guarantee these tires to last?
Problem 45
Companies that design furniture for elementary school classrooms produce a variety of sizes for kids of different ages. Suppose the heights of kindergarten children can be described by a Normal model with a mean of \(38.2\) inches and standard deviation of \(1.8\) inches. a) What fraction of kindergarten kids should the company expect to be less than 3 feet tall? b) In what height interval should the company expect to find the middle \(80 \%\) of kindergarteners? c) At least how tall are the biggest \(10 \%\) of kindergarteners?
Problem 48
. Agricultural scientists are working on developing an improved variety of Roma tomatoes. Marketing research indicates that customers are likely to bypass Romas that weigh less than 70 grams. The current variety of Roma plants produces fruit that averages 74 grams, but \(11 \%\) of the tomatoes are too small. It is reasonable to assume that a Normal model applies. a) What is the standard deviation of the weights of Romas now being grown? b) Scientists hope to reduce the frequency of undersized tomatoes to no more than \(4 \%\). One way to accomplish this is to raise the average size of the fruit. If the standard deviation remains the same, what target mean should they have as a goal? c) The researchers produce a new variety with a mean weight of 75 grams, which meets the \(4 \%\) goal. What is the standard deviation of the weights of these new Romas? d) Based on their standard deviations, compare the tomatoes produced by the two varieties.
