A hat shop owner is thinking about the number of hats to order from his supplier...

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A hat shop owner is thinking about the number of hats to order from his supplier for the following week. The cost of each hat is $7 and can be sold for $10. At the end of the week, any hats that are unsold will be sold at a 50% discount the following week. The demand distribution is as follows: 10 hats with probability of 12%, 15 hats with probability of 25%, 20 hats with probability of 30%, 25 hats with probability of 17%, 30 hats with probability of 16%. a) The hat shop owner decides to order 26 hats. What is his expected profit? b) What is the expected number of hats that the hat shop owner will have to sell at a discount next week? To ensure the quality of the hats sold at the shop, the circumference of each hat should be within the range of 50cm + 2cm. It was found that the hats' circumference is normally distributed with mean of 49cm and standard deviation of 1.8cm. c) What percentage of hats are acceptable at the shop? The hat shop also sells crystal jewelry. The hat shop owner claims that each of his crystal are approximately 8g each. The probability density function (pdf) of the crystals' weight is given by: 10 f(x) = 1, 3 d) What percentage of the crystals weigh less than 8.1g? e) What is the mean weight of the crystals? Calculate it using the expression of f(x). ,7.9 ≤ x ≤ 8.2 A hat shop owner is thinking about the number of hats to order from his supplier for the following week. The cost of each hat is $7 and can be sold for $10. At the end of the week, any hats that are unsold will be sold at a 50% discount the following week. The demand distribution is as follows: 10 hats with probability of 12%, 15 hats with probability of 25%, 20 hats with probability of 30%, 25 hats with probability of 17%, 30 hats with probability of 16%. a) The hat shop owner decides to order 26 hats. What is his expected profit? b) What is the expected number of hats that the hat shop owner will have to sell at a discount next week? To ensure the quality of the hats sold at the shop, the circumference of each hat should be within the range of 50cm + 2cm. It was found that the hats' circumference is normally distributed with mean of 49cm and standard deviation of 1.8cm. c) What percentage of hats are acceptable at the shop? The hat shop also sells crystal jewelry. The hat shop owner claims that each of his crystal are approximately 8g each. The probability density function (pdf) of the crystals' weight is given by: 10 f(x) = 1, 3 d) What percentage of the crystals weigh less than 8.1g? e) What is the mean weight of the crystals? Calculate it using the expression of f(x). ,7.9 ≤ x ≤ 8.2