Question

in a survey of 18 - year - old males, the mean weight was 154.9 pounds with a standard deviation of 45.4 pounds. assume the distribution can be approximated by a normal distribution.
(a) what weight represents the 94th percentile?
(b) what weight represents the 42nd percentile?
(c) what weight represents the third quartile?
(a) pounds
(round to one decimal place as needed.)

Explanation:

Step1: Recall the z - score formula

The formula for a value \(x\) in a normal distribution is \(x=\mu + z\sigma\), where \(\mu\) is the mean, \(z\) is the z - score, and \(\sigma\) is the standard deviation. Given \(\mu = 154.9\) pounds and \(\sigma=45.4\) pounds.

Step2: Find the z - score for the 94th percentile

Using a standard normal distribution table (or z - table), the z - score \(z\) corresponding to a cumulative probability of \(0.94\) is approximately \(z_{0.94}=1.55\).

Step3: Calculate the weight for the 94th percentile

Substitute \(\mu = 154.9\), \(z = 1.55\), and \(\sigma = 45.4\) into the formula \(x=\mu+z\sigma\). So \(x=154.9+1.55\times45.4\).
\[

$$\begin{align*} x&=154.9 + 1.55\times45.4\\ &=154.9+70.37\\ &=225.27\approx225.3 \end{align*}$$

\]

Step4: Find the z - score for the 42nd percentile

The cumulative probability is \(0.42\). Looking up in the z - table, the z - score \(z_{0.42}\approx - 0.20\).

Step5: Calculate the weight for the 42nd percentile

Substitute \(\mu = 154.9\), \(z=- 0.20\), and \(\sigma = 45.4\) into the formula \(x=\mu + z\sigma\). So \(x=154.9+( - 0.20)\times45.4\).
\[

$$\begin{align*} x&=154.9-0.20\times45.4\\ &=154.9 - 9.08\\ &=145.82\approx145.8 \end{align*}$$

\]

Step6: Recall the definition of the third - quartile

The third quartile \(Q_3\) represents the 75th percentile. The z - score \(z\) corresponding to a cumulative probability of \(0.75\) is approximately \(z_{0.75}=0.67\).

Step7: Calculate the weight for the third quartile

Substitute \(\mu = 154.9\), \(z = 0.67\), and \(\sigma = 45.4\) into the formula \(x=\mu+z\sigma\). So \(x=154.9+0.67\times45.4\).
\[

$$\begin{align*} x&=154.9+0.67\times45.4\\ &=154.9 + 30.418\\ &=185.318\approx185.3 \end{align*}$$

\]

Answer:

(a) 225.3 pounds
(b) 145.8 pounds
(c) 185.3 pounds