Question

Two students are interested in whether or not there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are more consistent.

What is the p-value?

Short answer

The$p-$value is$0.058$.

Step-by-step solution

Given Information

Two students are interested in whether or not there is variation in their test scores for math class. There are $15$ total math tests they have taken so far. The first student’s grades have a standard deviation of $38.1$. The second student’s grades have a standard deviation of $22.5$. The second student thinks his scores are more consistent.

Calculation

Let $1$and $2$be the subscripts that demonstrate the first and second student

$n_1=n_2=15$

A standard deviation for the first student is

$s_1=38.1$

A standard deviation for the second student is

$s_2=22.5$

$H_0:{\sigma }_1^2={\sigma }_2^2$and $H_a:{\sigma }_1^2<{\sigma }_2^2$

By the null hypothesis $\left({\sigma }_1^2={\sigma }_2^2\right)$, the $F$statistic is:

$F=\frac{\left[\frac{{\left(s_1\right)}^2}{{\left({\sigma }_1\right)}^2}\right]}{\left[\frac{{\left(s_1\right)}^2}{{\left({\sigma }_2\right)}^2}\right]}=\frac{{\left(s_1\right)}^2}{{\left(s_2\right)}^2}=\frac{38.1^2}{22.5^2}=2.867$.

Distribution for the test: $F_{14,14}$where $n_1-1=14$and $n_2-1=14$

Probability statement: $\mathrm{p}$-value$=\mathrm{P}(\mathrm{F}<2.867)=0.058$.