Mathematical Statistics and Probability

  
Exercise 1

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A consulting group believes that 70% of the people in a certain county are satisfied with their health coverage. Assuming that this is true, find the probability that in a random sample of 15 people from the county: (a) Exactly 10 are satisfied with their health coverage, and interpret. (b) Not more than 10 are satisfied with their health coverage, and interpret. (c) What is the expected number of people out of 15 that are satisfied with their health coverage?
Exercise 2
company that manufactures cars claims that the gas mileage for its new line of hybrid cars, on the average, is 60 miles per gallon with a standard deviation of 4 miles per gallon. A random sample of 16 cars yielded a mean of 57 miles per gallon. If the company’s claim is correct, what is the probability that the sample mean is less than or equal to 57 miles per gallon? Comment on the company’s claim about the mean gas mileage per gallon of its cars. What assumptions did you make?
Exercise 3
A random sample of size 50 from a particular brand of 16-ounce tea packets produced a mean weight of 15.65 ounces. Assume that the weights of these brands of tea packets are normally distributed with standard deviation of 0.59 ounce. Find a 95% confidence interval for the true mean μ.
Exercise 4
It is claimed that two of three Americans say that the chances of world peace are seriously threatened by the nuclear capabilities of other countries. If in a random sample of 400 Americans, it is found that only 252 hold this view, do you think the claim is correct? Use α = 0.05. State any assumptions you make in solving this problem.