Answer to Question #272350 in Statistics and Probability for Bin

Question

The average reading performance of grade 6 pupils was found to be 80 words per minute. A random sample of 20 students from three sections took the reading test and found to have an average speed of 74 words with a standard deviation of 8 words per minute. Can we say that the students performance is below the standard when tested at 0.05 level of significance?

Answer 1

Since the assumption that student's performance is below the standard is equivalent to assumption that population standard is below 80(in terms of testing), then

$H_0:a=80$

$H_1:a<80$ , where a - population mean

Since the population standard deviation is unknown and sample size is small then we should run t-test

t-statistic is calculated by next formula

$t={\frac {x-a} {\sigma}}*\sqrt{n}$ , where x - sample mean, a - supposed population mean, n - sample sizq, $\sigma$ - sample standard deviation

In the given case we have

$t={\frac {74-80} {8}}*\sqrt{20}=-3.35$

According to the form of alternative hypothesis, we will have enough statistical evidence to reject null hypothesis if t < Cr, where Cr is such value, that

$P(T(n-1)<Cr)=a=0.05$ , where T - Studnent's criteria with n - 1 degrees of freedom, n - sample size, a - level of significance. In the given case we have

$P(T(19)<Cr)=0.05\implies Сr = -1.73$

We receive that t < Cr, so we should reject the null hypothesis and admit that student's performance was indeed below standard at 0.05 level of significance