Question

The weight of a miniature Tootsie Roll is normally distributed with a mean of 3.30 grams and standard deviation of 0.13 grams.

(a) Within what weight range will the middle 95 percent of all
miniature Tootsie Rolls fall?
(b) What is the probability that a randomly chosen miniature
Tootsie Roll will weigh more than 3.50 grams?
c) What is the probability that a randomly chosen miniature
Tootsie Roll will weigh between 3.25 and 3.45 grams?

Answer 1

a) The middle 95 percent of all miniature Tootsie Rolls will fall between 3.04 grams and 3.56 grams.

b) 6.18% probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams.

c) 52.29% probability that a randomly chosen miniature Tootsie Roll will weigh between 3.25 and 3.45 grams

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.

The Empirical Rule is also used to solve this question. It states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

`\mu = 3.30, \sigma = 0.13`

(a) Within what weight range will the middle 95 percent of all miniature Tootsie Rolls fall?

By the Empirical Rule the weight range of the middle 95% of all miniature Tootsie Rolls fall within two standard deviations of the mean. So

3.30 - 2*0.13 = 3.04

3.30 + 2*0.13 = 3.56

The middle 95 percent of all miniature Tootsie Rolls will fall between 3.04 grams and 3.56 grams.

(b) What is the probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams?

This probability is 1 subtracted by the pvalue of Z when X = 3.50. So

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

`Z = (3.50 - 3.30)/(0.13)`

`Z = 1.54`

`Z = 1.54` has a pvalue of 0.9382.

1 - 0.9382 = 0.0618

6.18% probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams.

c) What is the probability that a randomly chosen miniature Tootsie Roll will weigh between 3.25 and 3.45 grams?

This is the pvalue of Z when X = 3.45 subtracted by the pvalue of Z when X = 3.25. So

X = 3.45

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

`Z = (3.45 - 3.30)/(0.13)`

`Z = 1.15`

`Z = 1.15` has a pvalue of 0.8749.

X = 3.25

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

`Z = (3.25 - 3.30)/(0.13)`

`Z = -0.38`

`Z = -0.38` has a pvalue of 0.3520

0.8749 - 0.3520 = 0.5229

52.29% probability that a randomly chosen miniature Tootsie Roll will weigh between 3.25 and 3.45 grams