Question

The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 330 grams and a standard deviation of 12 grams. Find the weight that corresponds to each event. (Use Excel or Appendix C to calculate the z-value. Round your final answers to 2 decimal places.) a. Highest 20 percent

Answer 1

Weights of at least 340.1 are in the highest 20%.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value 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. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 330, \sigma = 12`

a. Highest 20 percent

At least X

100-20 = 80

So X is the 80th percentile, which is X when Z has a pvalue of 0.8. So X when Z = 0.842.

`Z = (X - \mu)/(\sigma)`

`0.842 = (X - 330)/(12)`

`X - 330 = 12*0.842`

`X = 340.1`

Weights of at least 340.1 are in the highest 20%.