Answer to Question #284565 in Statistics and Probability for Knight

Question

The mean score of a group of 24 students in an educational achievement test in reading was 45 with a standard deviation of 6. At the end of the school year, the mean score on an equivalent form of the same test was 50 with a standard deviation of 5. Has the class made significant progress in reading during the year? Use alpha equal to 0.05.

Answer 1

Let $X_{1}$ denote the score at the beginning of the school year and $X_{2}$ denote the score at the end of the school year. let us assume that $x_{1} \sim N\left(\mu_{1}, \sigma_{1}{ }^{2}\right)$ and $x_{2} \sim N\left(\mu_{2}, \sigma_{2}{ }^{2}\right)$ independently. Assume that $\sigma_{1}{ }^{2}=\sigma_{2}{ }^{2}=\sigma^{2}$ (say)

We want to determine if the lass has made a significant progress in reading during the year. That is, the hypothesis is,

$H_{0}: \mu_{1}=\mu_{2} \quad r / s \quad H_{a}: \mu_{1}<\mu_{2}$

The appropriate teat statistic is,

$T=\frac{\bar{X}_{1}-\bar{X}_{2}}{s \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}} \sim_{H_o} t_{n_{1}+n_{2}-2}$

where $\bar{X}_{1}$ and $\bar{X}_{2}$ are the sample mean and s denote the pooled s.d.

Given,

$\begin{aligned} \bar{X}_{1} &=45, \bar{X}_{2}=50, n_{1}=24, n_{2}=24 \\ s^{2} &=\frac{\left(n_{1}-1\right) s_{1}^{2}+\left(n_{2}-1\right) A_{2}^{2}}{n_{1}+n_{2}-2}=\frac{23 \times 36+23 \times 25}{46} \\ &=30.5 \end{aligned}$

$\begin{aligned} \therefore s &=5.5 \\ \therefore t &=\frac{45-50}{5.5 \sqrt{\frac{2}{24}}}=\frac{-5}{1.587}=-3.149 \end{aligned}$

where t is the observed value of $T \sim t_{46}$

We reject $H_{0}$ at $\alpha$ level of significance, if, $t<-t_{\alpha ; 46} \quad$ [$\because$ the test is left tailed ]

$\begin{aligned} &\text { For } \alpha=0.05, \\ &-t_{\alpha ; 46}=-t_{0.05 ; 46}=-1.678 \\ &\therefore t=-3.149<-t_{0.05 ; 46} \end{aligned}$

$\therefore$ we reject $H_{0}$ and conclude that there in sufficient evidence to say that the class made a significant progress in reading during the year.