The weight of a product is normally distributed with a μ of 4oz and a σ of 0.55oz.

a) What is the probability that a randomly selected unit from a recently manufactured batch weighs more than 3.75 ounces?

b) The company wants to classify a unit as a scrap in a maximum of 5% of the units if the weight is below a desired value. Determine the desired weight such that no more than 5% of the units are below it.

c) What is the probability that an average of 36 of these products weighs more than 4.5 ounces?

d) A new issue has arisen and the products will need to be thrown out if they weigh too much. The goal of the production manager now is to re-set the machines to a new mean (not the original 4 ounces) so that no more than 10% of the products to weigh more than 5.1 ounces. What should the new average weight be?

 

A) 0.6736

B) x = 3.09525

C) 0

D) 4.396

Step-by-step explanation

The weight of a product is normally distributed with a μ of 4oz and a σ of 0.55oz.

a) What is the probability that a randomly selected unit from a recently manufactured batch weighs more than 3.75 ounces?

Use the equation:

z = (x-mean)/standard deviation

Plug in the mean and standard deviation, and use x as 3.75.

z = (3.75 - 4)/0.55

z = -0.25/0.55 = -0.45

Now, look up this value in the standard normal table to find the associated p-value.

p = 0.3264 when z = -0.45

This is the probability that your selection is less than 3.75. Subtract from 1 to find the probability that it is greater.

1-0.3264 = 0.6736

b) The company wants to classify a unit as a scrap in a maximum of 5% of the units if the weight is below a desired value. Determine the desired weight such that no more than 5% of the units are below it.

Here, you want to work backwards and start with p = 0.05. Look up z when p= 0.05 in the standard normal table. When p = 0.05, z = -1.645.

Now, plug this into the equation from part a with the mean and standard deviation.

-1.645 = (x - 4)/.55

-0.90475 = x - 4

x = 3.09525

c) What is the probability that an average of 36 of these products weighs more than 4.5 ounces?

To solve this problem, use the equation

z = (x - mean)/(standard deviation/sqrt(n))

N is the sample size, which here is 36.

Plug in the values.

z = (4.5 - 4)/(0.55/sqrt(36))

z = 0.5/(0.55/6)

z = 0.5/0.09167

z = 5.45

Look this up in the standard normal table.

When z = 5.45, p = >0.9999 (Round to 1)

This is the probability that the mean weight is less than 4.5. To find the probability it is greater, subtract this value from 1.

1 - 1 = 0

The probability that the mean weight is greater than 4.5 is so small that it is essentially 0.

d) A new issue has arisen and the products will need to be thrown out if they weigh too much. The goal of the production manager now is to re-set the machines to a new mean (not the original 4 ounces) so that no more than 10% of the products to weigh more than 5.1 ounces. What should the new average weight be?

Here, you want no more than 10% of products to weigh more than 5.1 ounces. Thus, we want to find z when p = 0.9, as this p value separates the smallest 90% of values from the 10% that are the largest. Look up z when p = 0.9.

z = 1.28 when p = 0.9

Plug this into the equation given in part a, where x = 5.1 and you solve for the mean.

1.28 = (5.1 - mean)/0.55

0.704 = 5.1 - mean

mean = 5.1 - 0.704

mean = 4.396