Question

33. Suppose that the scores on a statewide standardized test are normally distributed with a mean of 78 and a standard deviation of 3. Estimate the percentage of scores that were (a) between 75 and 81. % (b) above 87. % (c) below 72. % (d) between 75 and 84. %

Answer 1

Given the scores on a statewide standardized test are normally distributed

Mean = μ = 78

Standard deviation = σ = 3

Normalize the data using the z-score by using the following formula and chart:

$z=\frac{x-\mu}{\sigma}$

Estimate the percentage of scores of the following cases:

(a) between 75 and 81

so, the z-score for the given numbers will be:

$\begin{gathered} 75\rightarrow z=\frac{75-78}{3}=\frac{-3}{3}=-1 \\ 81\rightarrow z=\frac{81-78}{3}=\frac{3}{3}=1 \end{gathered}$

As shown, the percentage when (-1 < z < 1) = 68%

(b) above 87

$87\rightarrow z=\frac{87-78}{3}=\frac{9}{3}=3$

The percentage when (z > 3) = 0.5%

(c) below 72

$72\rightarrow z=\frac{72-78}{3}=\frac{-6}{3}=-2$

The percentage when (z < -2) = 0.5 + 2 = 2.5%

(d) between 75 and 84

$\begin{gathered} 75\rightarrow z=\frac{75-78}{3}=-\frac{3}{3}=-1 \\ 84\rightarrow z=\frac{84-78}{3}=\frac{6}{3}=2 \end{gathered}$

The percentage when ( -1 < z < 2 ) = 68 + 13.5 = 81.5%