Question

question 4
test scores are normally distributed with a mean of 100 and a standard deviation of 15. approximately what proportion of test-takers score more than 130?
a. 0.62%
b. 2.28%
c. 15.87%
d. 34.13%
question 5
the weights of adult cats in a population are normally distributed with a mean of 10 pounds and a standard deviation of 1.5 pounds. what is the approximate proportion of cats that weigh less than 7 pounds?
a. 0.15%
b. 2.28%
c. 15.87%
d. 34.13%
question 6
heights of adult males are normally distributed with a mean of 70 inches and a standard deviation of 2.5 inches. approximately what proportion of adult males are either shorter than 65 inches or taller than 75 inches?
a. 0.3%
b. 4.6%
c. 5.0%
d. 13.6%

Explanation:

Question 4

Step1: Calculate z-score

$z = \frac{X-\mu}{\sigma} = \frac{130-100}{15} = 2$

Step2: Find upper tail proportion

For $z=2$, the area above $z$ is $2.28\%$ (per empirical rule: 95% of data is within $\mu\pm2\sigma$, so 5% outside, half above $2\sigma$)

Question 5

Step1: Calculate z-score

$z = \frac{X-\mu}{\sigma} = \frac{7-10}{1.5} = -2$

Step2: Find lower tail proportion

For $z=-2$, the area below $z$ is $2.28\%$ (per empirical rule: 95% of data is within $\mu\pm2\sigma$, so 5% outside, half below $-2\sigma$)

Question 6

Step1: Calculate z-scores

$z_1 = \frac{65-70}{2.5} = -2$, $z_2 = \frac{75-70}{2.5} = 2$

Step2: Find total tail proportion

Area below $z=-2$ plus area above $z=2$ is $2.28\% + 2.28\% = 4.56\% \approx 4.6\%$

Answer:

Question 4: B. 2.28%
Question 5: B. 2.28%
Question 6: B. 4.6%