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

Answer:

Population Mean = 2.0

Population Standard deviation = 0.03

Step-by-step 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, $\mu$ = $\frac{1.9+2.1}{2}$ = 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 ;  

                $Z = \frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

              -1.6449 = $\frac{1.9 - 2.0}{\frac{\sigma}{\sqrt{n} } }$     ⇒  $\frac{\sigma}{\sqrt{n} } = \frac{-0.1}{-1.6449}$  

                                           ⇒ $\sigma =$ 0.0608 * $\sqrt{30}$  {as sample size is given 30}

                                           ⇒ $\sigma$ = 0.03 .

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