UCLA Machine Shop Manufacture

7.3-1. A machine shop manufactures toggle levers. A lever is flawed if a standard nut cannot be screwed onto the threads. Let p equal the proportion of flawed toggle levers that the shop manufactures. If there were 24 flawed levers out of a sample of 642 that were selected randomly from the production line, (a) Give a point estimate of p. (b) Use Equation 7.3-2 to find an approximate 95% confidence interval for p.

(c) Use Equation 7.3-4 to find an approximate 95% confidence interval for p.

(d) Use Equation 7.3-5 to find an approximate 95% confidence interval for p.

(e) Find a one-sided approximate 95% confidence interval for p that provides an upper bound for p.

7.3-2. Let p equal the proportion of letters mailed in the Netherlands that are delivered the next day. Suppose that y = 142 out of a random sample of n = 200 letters were delivered the day after they were mailed. (a) Give a point estimate of p. (b) Use Equation 7.3-2 to find an approximate 90% confidence interval for p.

(c) Use Equation 7.3-4 to find an approximate 90% confidence interval for p.

(d) Use Equation 7.3-5 to find an approximate 90% confidence interval for p.

(e) Find a one-sided approximate 90% confidence interval for p that provides a lower bound for p.

7.3-6. Let p equal the proportion of Americanswho select jogging as one of their recreational activities. If 1497 out of a random sample of 5757 selected jogging, find an approximate 98% confidence interval for p.

7.4-1. Let X equal the tarsus length for a male grackle. (The tarsus is part of a bird’s leg between what appears to be a backward-facing “knee” and what appears to be an “ankle” .) Assume that the distribution of X is N(μ, 4.84). Find the sample size n that is needed so that we are 95% confident that the maximum error of the estimate of μ is 0.4.

7.4-2. Let X equal the excess weight of soap in a “1000-gram” bottle. Assume that the distribution of X is N(μ, 169). What sample size is required so that we have 95% confidence that the maximum error of the estimate of μ is 1.5?

7.4-3. A company packages powdered soap in “6-pound” boxes. The sample mean and standard deviation of the soap in these boxes are currently 6.09 pounds and 0.02 pound, respectively. If themean fill can be lowered by 0.01 pound, $14,000 would be saved per year. Adjustments were made in the filling equipment, but it can be assumed that the standard deviation remains unchanged. (a) How large a sample is needed so that the maximum error of the estimate of the new μ is ε = 0.001 with 90% confidence?

(b) A random sample of size n = 1219 yielded x = 6.048 and s = 0.022. Calculate an approximate 90% confidence interval for μ.

(c) Estimate the savings per year with these new adjust-ments.

(d) Estimate the proportion of boxes that will now weigh less than six pounds.

7.4-9. A die has been loaded to slightly change the prob-ability of rolling a six. In order to estimate p, the new probability of rolling a six, how many times must the die be rolled so that we are approximately 99% confident that themaximumerrorof theestimateof p is ε = 0.02?

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