Answer to Question #323796 in Statistics and Probability for secret

Question

On the Standford-Binet test, the mean IQ is 100. A class of 21 kindergarten pupils were tested with a resulting mean of 105 and a standard deviation of 5.44. Does this group have a mean significantly different than the norm group? Use 0.01 as your level of significance.

Answer 1

$\mu=100, \ n=21, \ \bar{x}=105, \ s=5.44,\ \alpha=0.01.$

The null and alternative hypotheses are

$H_0:\mu=100,\\ H_1:\mu\neq100.$

Because $\sigma$ is unknown and he population is normally distributed, we use the t-test.

The test is a two-tailed test, the level of significance is $\alpha=0.01$ , and the degrees of freedom are

d.f. = 21 - 1 = 20. So, using t-table, the critical values are -t₀ = -2.845 and t₀ = 2.845. The rejection regions are t < -2.845 and t > 2.845. The standardized test statistic is

$t=\cfrac{\bar{x}-\mu}{s/\sqrt{n}}=\cfrac{105-100}{5.44/\sqrt{21}}=4.212>2.845.$

Because t is in the rejection region, we reject the null hypothesis, this group has a mean significantly different than the norm group.