Question

Improving SAT scores. Refer to the Chance(Winter2001) examination of Scholastic Assessment Test (SAT)scores of students who pay a private tutor to help them improve their results, Exercise 2.88 (p. 113). On the SAT—Mathematics test, these students had a mean score change of +19 points, with a standard deviation of 65 points. In a random sample of 100 students who pay a private tutor to help them improve their results, what is the likelihood that the sample mean score change is less than 10 points?

Short answer

The likelihood that the sample means score change is less than 10 points is 0.0838.

Step-by-step solution

Given information

Referring to exercise 2.88 (p.113), there is a study about 100 students who pay a private tutor to help them improve their results. The students had a mean score of 19 and a standard deviation of 65 points.

Determine the likelihood

Let’s consider the sample mean of ${\mu }_{\overline{x}}=19$.

And the sample standard deviation of

$\begin{array}{rcl}{\sigma }_{\overline{x}}& =& \frac{\sigma }{\sqrt{n}}\\ & =& \frac{65}{\sqrt{100}}\\ & =& 6.5\\ & & \end{array}$

So, the likelihood that the sample mean score change is less than 10 points is,

$\begin{array}{rcl}\mathrm{Pr}\left(\overline{X}<10\right)& =& \mathrm{Pr}\left(\frac{\overline{X}-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}<\frac{10-19}{6.5}\right)\\ & =& \mathrm{Pr}\left(z<-1.38\right)\\ & =& 0.0838\\ & & \end{array}$

Therefore, the likelihood is 0.0838.