Question

ACT scores have a mean of 20.8 and 9 percent of the scores are above 28. The scores have a distribution that is approximately normal. Find the standard deviation.

Answer 1

Answer: 5.37

Step-by-step explanation:

Let x =  ACT scores.

Given: ACT scores have a mean of 20.8 and 9 percent of the scores are above 28. The scores have a distribution that is approximately normal.

i.e. P(X>28)=0.09   (i)

Now,

$P(X>28)=P(\dfrac{X-\mu}{\sigma}>\dfrac{28-20.8}{\sigma})\\\\= P(z>\dfrac{7.2}{\sigma})\ \ \ \ [z=\dfrac{X-\mu}{\sigma}]$    (ii)

One -tailed z value for p-value of 0.09 =1.3408 [By z-table]

From (i) and (ii)

$\dfrac{7.2}{\sigma}=1.3408\\\\\Rightarrow\ \sigma=\dfrac{7.2}{1.3408}\\\\\Rightarrow\ \sigma=5.37$

Hence, the standard deviation = 5.37