Questions: The accompanying data represent the weights (in grams) of a random sample of 48 M 8 M plain candies. Complete parts (a) through (f). (a) Determine the sample standard deviation weight. 0.036 gram(s) (Round to three decimal places as needed.) (b) On the basis of the accompanying histogram, comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of M8Ms. A. The histogram is approximately bell-shaped so the Empirical Rule can be used. B. The histogram is not approximately bell-shaped so the Empirical Rule cannot be used. C. The histogram is approximately bell-shaped so the Empirical Rule cannot be used. D. The histogram is not approximately bell-shaped so the Empirical Rule can be used. (c) Use the Empirical Rule to determine the percentage of M8Ms with weights between 0.802 and 0.946 gram. Hint: x̄=0.874. % (Type an integer or decimal. Do not round.)

The accompanying data represent the weights (in grams) of a random sample of 48 M 8 M plain candies. Complete parts (a) through (f).
(a) Determine the sample standard deviation weight.
0.036 gram(s)
(Round to three decimal places as needed.)
(b) On the basis of the accompanying histogram, comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of M8Ms.
A. The histogram is approximately bell-shaped so the Empirical Rule can be used.
B. The histogram is not approximately bell-shaped so the Empirical Rule cannot be used.
C. The histogram is approximately bell-shaped so the Empirical Rule cannot be used.
D. The histogram is not approximately bell-shaped so the Empirical Rule can be used.
(c) Use the Empirical Rule to determine the percentage of M8Ms with weights between 0.802 and 0.946 gram. Hint: x̄=0.874.
% (Type an integer or decimal. Do not round.)

Transcript text: The accompanying data represent the weights (in grams) of a random sample of 48 M 8 M plain candies. Complete parts (a) through (f). (a) Determine the sample standard deviation weight. $0.036 \mathrm{gram}(\mathrm{s})$ (Round to three decimal places as needed.) (b) On the basis of the accompanying histogram, comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of M8Ms. A. The histogram is approximately bell-shaped so the Empirical Rule can be used. B. The histogram is not approximately bell-shaped so the Empirical Rule cannot be used. C. The histogram is approximately bell-shaped so the Empirical Rule cannot be used. D. The histogram is not approximately bell-shaped so the Empirical Rule can be used. (c) Use the Empirical Rule to determine the percentage of M8Ms with weights between 0.802 and 0.946 gram. Hint: $\bar{x}=0.874$. $\square$ \% (Type an integer or decimal. Do not round.)

Solution

Solution Steps

Step 1: Calculate the Mean

The mean $ \mu $ of the weights is calculated as follows: \[ \mu = \frac{\sum x_i}{n} = \frac{41.949999999999996}{48} = 0.874 \]

Step 2: Calculate the Variance

The variance $ \sigma^2 $ is determined using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 0.001 \]

Step 3: Calculate the Standard Deviation

The sample standard deviation $ \sigma $ is the square root of the variance: \[ \sigma = \sqrt{0.001} = 0.036 \]

Step 4: Assess the Appropriateness of the Empirical Rule

Based on the shape of the histogram, we conclude: A. The histogram is approximately bell-shaped so the Empirical Rule can be used.

Step 5: Calculate the Z-Scores

The z-scores for the weights 0.802 and 0.946 grams are calculated as follows: \[ z = \frac{X - \mu}{\sigma} = \frac{0.802 - 0.874}{0.036} = -2.0 \] \[ z = \frac{X - \mu}{\sigma} = \frac{0.946 - 0.874}{0.036} = 2.0 \]

Step 6: Determine the Percentage of M&Ms within the Weight Range

Using the Empirical Rule, we find the percentage of M&Ms with weights between 0.802 and 0.946 grams: \[ \text{Percentage} = 95.45\% \]