Question

Sarah's composite ACT score was in the 85th percentile. If all composite ACT scores are normally distributed with a meanof 18 and a standard deviation of 6, what is Sarah's composite score? Round your answer to the nearest whole number.

Answer 1

24.24

Step-by-step explanation:

Sarah's composite score is the score such that:

P(X≤x) = 0.85

It means that the probability that a score will be lower than Sarah's score is 0.85.

But we have the standardized normal distribution table, so we first, will find a z such that:

P(Z≤z) = 0.85

Using the table, the closest number to 0.85 is 0.8508, so the value of z is 1.04.

1.0 from the row added to 0.04 from the column.

Now, x and z are related as:

`z=(x-m)/(s)`

Where m is the mean and s is the standard deviation. So, replacing z by 1.04, m by 18, and s by 6, we get:

`1.04=(x-18)/(6)`

So, solving for x, we get:

`\begin{gathered} 1.04\cdot6=(x-18)/(6)\cdot6 \\ 6.24=x-18 \\ 6.24+18=x-18+18 \\ 24.24=x \end{gathered}`

Therefore, Sarah's composite score is 24.24