Question

A production process is checked periodically by a quality control inspector. the inspector selects simple random samples of 30 finished products and computes the sample mean product weight. If test results over a long period of time show that 5% of the values are over 2.1 pounds and 5% are under 1.9 pounds, what are the mean and the standard deviation for the population of products produced with this process? What is the population mean? (to 1 decimal) What is the population standard deviation (to 2 decimals)?

Answer 1

Population Mean = 2.0

Population Standard deviation = 0.03

Explanation:

We are given that the inspector selects simple random samples of 30 finished products and computes the sample mean product weight.

Also, test results over a long period of time show that 5% of the values are over 2.1 pounds and 5% are under 1.9 pounds.

Now, mean of the population is given the average of two extreme boundaries because mean lies exactly in the middle of the distribution.

So, Mean,
`\mu` =
`(1.9+2.1)/(2)` = 2.0

Therefore, mean for the population of products produced with this process is 2.

Since, we are given that 5% of the values are under 1.9 pounds so we will calculate the z score value corresponding to a probability of 5% i.e.

z = -1.6449 {from z % table}

We know that z formula is given by ;

`Z = (Xbar - \mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

-1.6449 =
`(1.9 - 2.0)/((\sigma)/(√(n) ) )` ⇒
`(\sigma)/(√(n) ) = (-0.1)/(-1.6449)`

⇒
`\sigma =` 0.0608 *
`√(30)` {as sample size is given 30}

⇒
`\sigma` = 0.03 .

Therefore, Standard deviation for the population of products produced with this process is 0.0333.