Question

A production process is checked periodically by a quality control inspector. The inspector selects simple random samples of 50 finished products and computes the sample mean product weights x. If test results over a long period of time show that 5% of the x values are over 4.1 pounds and 5% are under 3.9 pounds, what is the standard deviation (in lb) for the population of products produced with this process? (Round your answer for the standard deviation to two decimal places.)

Answer 1

Let X be the product weight of a randomly selected product from the population of products.

Let the mean weight for the population of products (that is mean of X) be and the standard deviation of weight for the population of products (standard deviation of X) be .

Let be the sample mean weights for a simple random samples of 50 finished products.

Using the central limit theorem, we know that since the sample size is greater than 30, is normally distributed with mean and standard deviation (or standard error of mean)

We have that 5% of the values are over 4.1 pounds. That is the probability that is greater than 4.1 pounds is 0.05

Next, we know that 5% of the values are under 3.9 pounds. That is the probability that is less than 3.9 pounds is 0.05

Now we need to find the value of Z for which P(Z<z) = 0.05. We know that the area under the standard normal curve to the left of mean (which is zero) is 0.5. That is P(Z<0) = 0.5. Since 0.05 is less than 0.5, the value of z must be negative.

Hence we need P(Z<-z) = 0.05. This is same as

From the standard normal tables we know that P(Z<1.645) = 0.95. Hence P(Z<-1.645) = 0.05

Now we equate the z score of 3.9 to -1.645 and get

By equating the 2 values of mu we get

ans: The standard deviation (in lb) for the population of products produced with this process is 0.43 lbs