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=(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=(75-78)/(3)=(-3)/(3)=-1 \\ 81\rightarrow z=(81-78)/(3)=(3)/(3)=1 \end{gathered}`

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

(b) above 87

`87\rightarrow z=(87-78)/(3)=(9)/(3)=3`

The percentage when (z > 3) = 0.5%

(c) below 72

`72\rightarrow z=(72-78)/(3)=(-6)/(3)=-2`

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

(d) between 75 and 84

`\begin{gathered} 75\rightarrow z=(75-78)/(3)=-(3)/(3)=-1 \\ 84\rightarrow z=(84-78)/(3)=(6)/(3)=2 \end{gathered}`

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