91Ó°ÊÓ

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.

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

$s_2=22.5$

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

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

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

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

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