select the correct answer. you have a list of 600 random, normally distributed numbers with a mean of 10 and…

select the correct answer. you have a list of 600 random, normally distributed numbers with a mean of 10 and a standard deviation of 2. which of these statements is true? a. about 489 numbers would lie between the values 8 and 12. b. about 408 numbers would lie between the values 6 and 10. c. about 285 numbers would lie between the values 10 and 14. d. exactly 408 numbers would lie between the values 10 and 14. e. exactly 285 numbers would lie between the values 8 and 12.

select the correct answer. you have a list of 600 random, normally distributed numbers with a mean of 10 and a standard deviation of 2. which of these statements is true? a. about 489 numbers would lie between the values 8 and 12. b. about 408 numbers would lie between the values 6 and 10. c. about 285 numbers would lie between the values 10 and 14. d. exactly 408 numbers would lie between the values 10 and 14. e. exactly 285 numbers would lie between the values 8 and 12.

Answer

Explanation:

Step1: Recall the empirical rule for normal distribution

For a normal - distribution, about 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard deviations of the mean ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard deviations of the mean ($\mu\pm3\sigma$). Given $\mu = 10$ and $\sigma=2$.

Step2: Analyze the range 8 - 12

The range 8 - 12 is $\mu-\sigma$ to $\mu + \sigma$. Since about 68% of the data lies within this range, and we have $n = 600$ data - points. The number of data - points in this range is $0.68\times600=408$.

Step3: Analyze the range 6 - 10

The range 6 - 10 is $\mu - 2\sigma$ to $\mu$. Since about 95% of the data lies within $\mu\pm2\sigma$, the proportion of data between $\mu - 2\sigma$ and $\mu$ is $\frac{0.95}{2}=0.475$. The number of data - points in this range is $0.475\times600 = 285$.

Step4: Analyze the range 10 - 14

The range 10 - 14 is $\mu$ to $\mu+2\sigma$. The proportion of data between $\mu$ and $\mu + 2\sigma$ is $\frac{0.95}{2}=0.475$. The number of data - points in this range is $0.475\times600=285$.

Answer:

A. About 408 numbers would lie between the values 8 and 12.