Question

Potato chips The distribution of weights of 9-ounce bags of a particular brand of potato chips is approximately Normal with a mean $\mu =9.12$ounces and standard deviation $\alpha =0.05$ounce. Draw an accurate sketch of the distribution of potato chip bag weights. Be sure to label the mean, as well as the points one, two, and three standard deviations away from the mean on the horizontal axis. Use rule$68-95-99.7$.

a) What percent of bags weigh less than $9.02$ounces?

(b) Between what weights do the middle $68\%$of bags fall?

(c) What percent of 9-ounce bags of this brand of potato chips weigh between $8.97$and $9.17$ounces?

(d) A bag that weighs$9.07$ ounces is at what percentile in this distribution?

Short answer

(a) About $2.5\%$of bags weigh less than $9.02$ounces.

(b) Roughly $68\%$of bags had a weight of $9.07$ounces to $9.17$ounces.

(c) $83.85\%$of the bags weigh between $8.97$and $9.17$ounces.

(d) The $16$th percentile of the weights of these potato chips packages is $9.07.$

Step-by-step solution

Part (a) Step 1: Given information

A particular brand of potato chips is approximately Normal with

The mean is$9.12$

The standard deviation is$0.05$

Part (a) Step 2: Explanation

The weight distribution is approximately $N(9.12,0.05)$. The Normal density curve is sketched below, with the necessary spots labelled.

By the $68-95-99.7$rule, approximately $2.5\%$of bags weigh less than $9.02$ounces, because $9.02$is two standard deviations below the mean.

As a result, about $2.5\%$of bags weigh less than $9.02$ounces.

Part (b) Step 1: Given information

A particular brand of potato chips is approximately Normal with

The mean is $9.12$

Standard deviation is$0.05$

Part (b) Step 2: Explanation

Approximately $68\%$of bags had weights between $9.12-0.05=9.07$and $9.12+0.05=9.17$ounces, according to the $68-95-99.7$standards.

As a result, roughly $68\%$of bags had a weight of $9.07$ounces to $9.17$ounces.

Part (c) Step 1: Given information

A particular brand of potato chips is approximately Normal with

The mean is $9.12$

Standard deviation is$0.05$

Part (c) Step 2: Explanation

Because $9.17$is one standard deviation above the mean, according to the $68-95-99.7$rule, about $84\%(0.68+0.16)$of bags have weights less than $9.17$ounces. Because $8.97$is three standard deviations below the mean, approximately $0.15\%$of bags weigh less than $8.97$ounces. As a result, about $84\%-0.15\%=83.85\%$of bags weigh between $8.97$and $9.17$ounces.

As a result, $83.385\%$of the bags weigh between $8.97$and $9.17$ounces.

Part (d) Step 1: Given information

A particular brand of potato chips is approximately Normal with

The mean is $9.12$

Standard deviation is$0.05$.

Part (d) Step 2: Explanation

Then $9.07$is one standard deviation below the average.

The area to the left of $9.07$is therefore $0.16$. In other words, the $16$th percentile of the weights of these potato chips packages is$9.07$.