Question

the heights of 3 - year - old females are normally distributed with a mean of 38 inches and a standard deviation of 1.8 inches. which of the following is the best estimate of the 84th percentile?
40.7 inches
39.8 inches
38 inches
41.6 inches
49.8 inches

Explanation:

Step1: Recall the properties of the normal distribution

In a normal distribution, about 68% of the data lies within \( \mu\pm\sigma \), 95% within \( \mu\pm2\sigma \), and 99.7% within \( \mu\pm3\sigma \). The 50th percentile is the mean \( \mu \). The 84th percentile is approximately \( \mu + \sigma \) because \( P(X\leq\mu+\sigma)\approx0.84 \) (since \( P(\mu - \sigma<X<\mu+\sigma)\approx0.68 \), so \( P(X\leq\mu+\sigma)=P(X\leq\mu)+\frac{P(\mu - \sigma<X<\mu+\sigma)}{2}=0.5 + 0.34=0.84 \))

Step2: Substitute the values of \( \mu \) and \( \sigma \)

Given \( \mu = 38 \) inches (mean) and \( \sigma=1.8 \) inches (standard deviation).
Using the formula \( x=\mu+\sigma \), we substitute the values: \( x = 38+1.8\)

Step3: Calculate the value

\( x=39.8 \) inches

Answer:

39.8 inches