Question

7.32 the weight of a miniature tootsie roll is normally distributed with a mean of 3.30 grams and standard deviation of 0.13 gram. (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? (data are from a project by mba student henry scussel.)

Explanation:

Step1: Recall the empirical rule for normal distribution

For a normal - distribution, about 95% of the data lies within 2 standard deviations of the mean.
Let $\mu = 3.30$ grams and $\sigma=0.13$ grams.
The lower bound of the range is $\mu - 2\sigma$ and the upper bound is $\mu + 2\sigma$.

Step2: Calculate the lower bound

$\mu - 2\sigma=3.30-2\times0.13=3.30 - 0.26 = 3.04$ grams.

Step3: Calculate the upper bound

$\mu + 2\sigma=3.30 + 2\times0.13=3.30+0.26 = 3.56$ grams.

Step4: Standardize the value for part (b)

We use the z - score formula $z=\frac{x-\mu}{\sigma}$, where $x = 3.50$, $\mu = 3.30$ and $\sigma = 0.13$.
$z=\frac{3.50 - 3.30}{0.13}=\frac{0.20}{0.13}\approx1.54$.

Step5: Find the probability

We want $P(X>3.50)$. Using the standard normal distribution table, $P(Z > 1.54)=1 - P(Z\leq1.54)$.
From the standard - normal table, $P(Z\leq1.54)=0.9382$. So $P(Z > 1.54)=1 - 0.9382 = 0.0618$.

Answer:

(a) The middle 95 percent of all miniature Tootsie Rolls will fall within the weight range of 3.04 grams to 3.56 grams.
(b) The probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams is 0.0618.