Final answer:
The store can expect to sell between 70 and 85 copies of the magazine approximately 76% of the weeks. The probability of selling less than 66 magazines is approximately 6.7%. The percentage of weeks with an insufficient number of copies to meet demand is approximately 3.3% given that the store stocks 86 copies.
Step-by-step explanation:
To answer part a), we need to find the probability that the store sells between 70 and 85 copies of the magazine. We can do this by finding the area under the Normal curve between the z-scores corresponding to 70 and 85.
First, we need to convert the values of 70 and 85 into z-scores. To do this, we use the formula z = (x - mean) / standard deviation. Plugging in the values for mean and standard deviation, we get z = (70 - 75) / 6 = -0.83, and z = (85 - 75) / 6 = 1.67.
Next, we use a Normal distribution table or a calculator to find the area under the curve between these two z-scores. The percentage of weeks the store can expect to sell between 70 and 85 copies is equal to the area under the curve, which corresponds to a probability of approximately 76%.
To answer part b), we need to find the probability that the store sells less than 66 magazines. We can do this by finding the area under the curve to the left of the z-score corresponding to 66. Using the same formula as before, we find that the z-score for 66 is (66 - 75) / 6 = -1.5. Using a Normal distribution table or a calculator, we find that the area to the left of -1.5 is approximately 0.067, or 6.7%.
Finally, to answer part c), we need to find the percentage of weeks that the store will have an insufficient number of copies to meet demand, given that the store stocks 86 copies. We can find this by finding the probability that the store will sell more than 86 copies. Using the formula and calculations from part b), we find that the z-score for 86 is (86 - 75) / 6 = 1.83. Using a Normal distribution table or a calculator, we find that the area to the right of 1.83 is approximately 0.033, or 3.3%.