Question

The weight of a miniature Tootsie Roll is normally distributed with a mean of 3.30 grams and standard deviation of .13 gram

A. Within what weight range will the middle 95% of all miniature tootsie rolls fall hint use the empirical rule
B. What is the probability that a randomly chosen miniature tootsie roll will weigh more than 3.50 grams(round your answer to 4 decimal places)
Answer all questions please URGENT

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 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.

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 x 0.13 = 3.04

3.30 + 2 x 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 p-value 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 p-value 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 p-value of Z when X = 3.45 subtracted by the p-value 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 p-value 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 p-value 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.