Question

The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 325 grams and a standard deviation of

10 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.)
3378
318.26
to
33174
a. Highest 10 percent
b. Middle 50 percent
c. Highest 80 percent
d. Lowest 10 percent

Answer 1

a. Above 337.8 grams.

b. Between 318.25 grams and 331.75 grams.

c. Above 316.59 grams.

d. Below 312.2 grams

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean of 325 grams and a standard deviation of 10 grams.

This means that
`\mu = 325, \sigma = 10`

a. Highest 10 percent

This is X when Z has a pvalue of 1 - 0.1 = 0.9, so X when Z = 1.28.

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

`1.28 = (X - 325)/(10)`

`X - 325 = 10*1.28`

`X = 337.8`

So 337.8 grams.

b. Middle 50 percent

Between the 50 - (50/2) = 25th percentile and the 50 + (50/2) = 75th percentile.

25th percentile:

X when Z has a pvalue of 0.25, so X when Z = -0.675.

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

`-0.675 = (X - 325)/(10)`

`X - 325 = -0.675*10`

`X = 318.25`

75th percentile:

X when Z has a pvalue of 0.75, so X when Z = 0.675.

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

`0.675 = (X - 325)/(10)`

`X - 325 = 0.675*10`

`X = 331.75`

Between 318.25 grams and 331.75 grams.

c. Highest 80 percent

Above the 100 - 80 = 20th percentile, which is X when Z has a pvalue of 0.2. So X when Z = -0.841.

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

`-0.841 = (X - 325)/(10)`

`X - 325 = -0.841*10`

`X = 316.59`

Above 316.59 grams.

d. Lowest 10 percent

Below the 10th percentile, which is X when Z has a pvalue of 0.1, so X when Z = -1.28.

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

`-1.28 = (X - 325)/(10)`

`X - 325 = -1.28*10`

`X = 312.2`

Below 312.2 grams