Question

A local company makes snack-size bags of potato chips. Each day, the company produces batches of 400 snack-size bags using a process designed to fill each bag with an average of 2 ounces of potato chips. However, due to imperfect technology, the actual amount placed in a given bag varies. Assume the amount placed in each of the 400 bags is normally distributed and has a standard deviation of 0.1 ounce. What is the probability that a sample of 40 bags has an average weight of at least 2.02 ounces

Answer 1

Answer:

Probability that a sample of 40 bags has an average weight of at least 2.02 ounces is 0.103.

Step-by-step explanation:

We are given that the company produces batches of 400 snack-size bags using a process designed to fill each bag with an average of 2 ounces of potato chips. Assume the amount placed in each of the 400 bags is normally distributed and has a standard deviation of 0.1 ounce.

Also, sample of 40 bags are selected.

Let $\bar X$ = sample average weight

The z-score probability distribution for sample mean is given by ;

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

where, $\mu$ = population mean weight of potato chips = 2 ounces

            $\sigma$ = standard deviation = 0.1 ounces

            n = sample of bags = 40

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, the probability that a sample of 40 bags has an average weight of at least 2.02 ounces is given by = P($\bar X$ $\geq$ 2.02 ounces)

   P($\bar X$ $\geq$ 2.02) = P( $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ $\geq$ $\frac{2.02-2}{\frac{0.1}{\sqrt{40} } }$ ) = P(Z $\geq$ 1.265) = 1 - P(Z < 1.265)

                                                      = 1 - 0.89707 = 0.103

The above probability is calculated using z table by looking at value of x = 1.265 which will lie between x = 1.26 and x = 1.27 in the z table which have an area of 0.89707.

Therefore, probability that a sample of 40 bags has an average weight of at least 2.02 ounces is 0.103.