Posted: November 24th, 2022

Problem 1: A consumer advocate claims that 80 percent of cable television subscribers are not satisfied with their cable service. In an attempt to justify this claim, a randomly selected sample of 50 cable subscribers is selected. Assuming independence, find:

1) The probability that 30 or fewer subscribers in the sample are not satisfied with their service.

2) The probability the more than 40 subscribers in the sample are not satisfied with their service.

3) The probability that between 40 and 45 (inclusive) subscribers in the sample are not satisfied with their service.

4) The probability that exactly 25 subscribers in the sample are satisfied with their service.

5) Suppose that when we survey 50 randomly selected cable television subscribers, we find that 30 are not satisfied with their service. Using a probability you found in this problem as the basis for you answer, do you believe the consumer advocate’s claim? Explain in one or two sentences.

Problem 2: A consensus forecast is the average of a large number of individual analysts’ forecasts. Suppose the individual forecasts for a particular interest rate are normally distributed with a mean of 5.0 percent and standard deviation of 1.2 percent. A single analyst is randomly selected.

a. Find the probability that his/her forecast is:

1) At least 3.5 percent.

2) At most 6 percent.

3) Between 3.5 and 6 percent.

b. What percentage of individual forecasts are at or below the 10th percentile of the distribution of forecasts? What percentage are at or above the 10th percentile? Find the 10th percentile of the distribution of individual forecasts.

c. Find the first quartile, Q1 and the third quartile, Q3, of the distribution of individual forecasts.

Problem 3: Each day a manufacturing plant receives a large shipment of drums of Chemical ZX-900. These drums are supposed to have a mean fill of 50 gallons, while the fills have a standard deviation known to be 0.6 gallons.

a. Suppose that mean fill for the shipment is actually 50 gallons. If we draw a random sample of 100 drums from the shipment, what is the probability that the average fill for the 100 drums is between 49.88 gallons and 50.12 gallons?

b. The plant manager is worried that the drums of Chemical ZX-900 are undefiled. Because of this, she decides to draw a sample of 100 drums from each daily shipment and will reject the shipment (send it back to the supplier) if the average fill for the 100 drums is less than 49.85 gallons. Suppose that a shipment that actually has a mean fill of 50 gallons is received. What is the probability that this shipment will be rejected and sent back to the supplier? What percent of the time will this happen?

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