Answer :
Answer:
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
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 3.30, \sigma = 0.13[/tex]
(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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.50 - 3.30}{0.13}[/tex]
[tex]Z = 1.54[/tex]
[tex]Z = 1.54[/tex] 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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.45 - 3.30}{0.13}[/tex]
[tex]Z = 1.15[/tex]
[tex]Z = 1.15[/tex] has a pvalue of 0.8749.
X = 3.25
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.25 - 3.30}{0.13}[/tex]
[tex]Z = -0.38[/tex]
[tex]Z = -0.38[/tex] 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