Question

1. mens weights are normally distributed with a mean of 180 pounds and a standard deviation of 25 pounds. womens weights are normally distributed with a mean of 140 pounds and a standard deviation of 20 pounds.

a. a new male student walks into class. he weighs 240 pounds. what is the z score associated with his weight?
b. he is followed by a female student who weighs 200 pounds. what is her z score?
c. which of these weights is more extreme? explain.

Explanation:

Step1: Recall Z - score formula

The formula for the Z - score is $Z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean and $\sigma$ is the standard deviation.

Step2: Calculate male student's Z - score

For the male student, $\mu = 180$ (mean of men's weights), $\sigma=25$ (standard deviation of men's weights) and $x = 240$. Substitute these values into the formula: $Z_m=\frac{240 - 180}{25}=\frac{60}{25}=2.4$.

Step3: Calculate female student's Z - score

For the female student, $\mu = 140$ (mean of women's weights), $\sigma = 20$ (standard deviation of women's weights) and $x = 200$. Substitute into the formula: $Z_f=\frac{200-140}{20}=\frac{60}{20}=3$.

Step4: Determine more extreme weight

The magnitude of the Z - score represents how many standard deviations a value is from the mean. A larger magnitude of the Z - score indicates a more extreme value. Since $|Z_f| = 3$ and $|Z_m|=2.4$, and $3>2.4$, the female student's weight is more extreme.

Answer:

a. $2.4$
b. $3$
c. The female student's weight is more extreme because her Z - score of 3 has a larger magnitude than the male student's Z - score of 2.4.