I need this please

Question

. A bookstore has determined that weekly sales of a pop culture magazine can be modeled by an approximately Normal distribution with a mean of 75 copies and a standard deviation of 6 copies.

a. For what percentage of the weeks can the store expect to sell between 70 and 85 copies of the magazine?

b. For what percentage of the weeks can the store expect to sell less than 66 magazines?

c. If the bookstore stocks 86 copies of the magazine, what percentage of weeks will there be an insufficient number of copies to meet demand?

Answer 1

Let x be the number of magazines sold, which is normally distributed with mean 75 and standard deviation 6.

For part a. we want to find P(70 < x < 85).

We convert the above range to z scores and use standardized normal table to find probabilities.

P((70 - 75)/6 < z < (85-75)/6)

P(-0.833 < z < 1.67)

Since the standardized normal table provides cumulative probabilities to the left, the above is the same as

P(z < 1.67) - P(z<-0.833) = 0.9525 - 0.2024 = 0.7501

For part b we want to find P(X < 66). Again we convert to a z-score as for part a then look up the probability from the standardized normal table.

For part c, they want to know the probability 86 copies will be insufficient, so we want to determine P(X > 86). Again we convert to a z-score. However, since the standardized normal table provides cumulative probabilities to the left, we will compute the probability as 1 - P(z < z-score).