Question: (i) The weight of turkeys is known to be normally distributed with a mean of 22 pounds and a standard deviation of 5 pounds.

1. What is the probability that a randomly selected turkey has the weight less than 20 pounds?
2. A random sample of 16 turkeys is selected. What is the probability that the sample mean is more than 25 pounds?
3. What is the probability that the mean weight of a random sample of 25 turkeys is between 18 and 24 pounds?

(ii) The director of admissions at a university says that 15% of high school juniors to whom she sends university literature eventually apply for admission. In a sample of 300 persons to whom materials were sent, 30 students applied for admission.

1. Assume that the chairman is correct and p = 0.15. What is the sampling distribution of the sample proportion \(\hat{p}\). Explain.
2. Find the expected value and the standard error of the sampling distribution of \(\hat{p}\).
3. What is the probability that the sample proportion \(\hat{p}\) will be between 0.09 and 0.12?
4. What is the probability that the sample proportion \(\hat{p}\) will be within ±0.05 of the population proportion p?

(iii) A certain brand of refrigerators has a length of life that is normally distributed with a mean and standard deviation of 15 years and 2 years, respectively.

1. What is the probability that a refrigerator lasts less than 6.5 years?
2. What is the probability that a refrigerator lasts more than 23 years?
3. What is the probability that a refrigerator lasts between 5 and 10 years?
4. What length of life should the retailer advertise for these refrigerators so that only 3% of the refrigerators fail before the advertised length of time?
5. What is the probability that a refrigerator lasts exactly 5 years?

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