Question

in a survey of 18 - year - old males, the mean weight was 151.5 pounds with a standard deviation of 48.7 pounds. assume the distribution can be approximated by a normal distribution. (a) what weight represents the 98th percentile? (b) what weight represents the 37th percentile? (c) what weight represents the first quartile? (a) □ pounds (round to one decimal place as needed)

Explanation:

Step1: Recall z - score formula

For a normal distribution $X\sim N(\mu,\sigma^{2})$, the z - score is given by $z=\frac{x - \mu}{\sigma}$, where $x$ is the value from the original distribution, $\mu$ is the mean and $\sigma$ is the standard deviation. We need to find the z - score corresponding to the given percentiles and then solve for $x$. Here, $\mu = 151.5$ and $\sigma=48.7$.

Step2: Find z - score for 98th percentile

Using a standard normal distribution table (or z - table), the z - score $z$ corresponding to a cumulative probability of $0.98$ is approximately $z_{0.98}=2.05$.

Step3: Solve for $x$ for 98th percentile

Substitute into the z - score formula: $2.05=\frac{x - 151.5}{48.7}$. Then $x=151.5+2.05\times48.7$. Calculate $2.05\times48.7 = 2.05\times(48 + 0.7)=2.05\times48+2.05\times0.7=98.4+1.435 = 99.835$. So $x=151.5 + 99.835=251.335\approx251.3$.

Step4: Find z - score for 37th percentile

The z - score $z$ corresponding to a cumulative probability of $0.37$ is approximately $z_{0.37}=- 0.33$.

Step5: Solve for $x$ for 37th percentile

Substitute into the z - score formula: $-0.33=\frac{x - 151.5}{48.7}$. Then $x=151.5-0.33\times48.7$. Calculate $0.33\times48.7 = 0.33\times(48+0.7)=0.33\times48 + 0.33\times0.7=15.84+0.231 = 16.071$. So $x=151.5-16.071 = 135.429\approx135.4$.

Step6: Find z - score for first quartile

The first quartile is the 25th percentile. The z - score $z$ corresponding to a cumulative probability of $0.25$ is approximately $z_{0.25}=-0.67$.

Step7: Solve for $x$ for first quartile

Substitute into the z - score formula: $-0.67=\frac{x - 151.5}{48.7}$. Then $x=151.5-0.67\times48.7$. Calculate $0.67\times48.7=0.67\times(48 + 0.7)=0.67\times48+0.67\times0.7=32.16+0.469 = 32.629$. So $x=151.5-32.629 = 118.871\approx118.9$.

Answer:

(a) 251.3 pounds
(b) 135.4 pounds
(c) 118.9 pounds