Question

A college professor claims that the entering class this year appears to be smarter than entering classes from previous years. He tests a random sample of 22 this year's entering students and finds that their mean IQ score is 119, with standard deviation of 15. The college records indicate that the mean IQ score for entering students from previous years is 112. If we assume that the IQ scores of this year's entering class are normally distributed, is there enough evidence conclude, at the 0.05 level of significance, that the mean IQ score, H, of this year's class is greater than that of previous years? Perform a one-tailed test. Then fill in the table below. Carry your intermediate computations to at least three decimal places and round your answers as specified in the table. μ σ The null hypothesis: The alternative hypothesis: The type of test statistic: (Choose one) 0-0 Oso O-O OO 00 x 5 ? The value of the test statistic: (Round to at least three decimal places.) The critical value at the 0.05 level of significance: (Round to at least three doomal places.) Can we conclude, using the 0.05 level of significance that the mean I score of this year's daus is greater than that of previous years? Yes No

Answer 1

Here claim is that the mean IQ score, $\mu$, f this year's class is greater than that of previous years.

So null hypothesis is $H_0:\mu=112$

Alternative hypothesis is $H_1:\mu>112$

As n is less than 30 and population standard deviation is not know so we will use t distribution

$t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}=\frac{119-112}{\frac{15}{\sqrt{22}}}=2.19$

The t-critical value for a right-tailed test, for a significance level of α=0.05 is

tc​=1.721

Graphically

$normaldistributiongrapher.php?&mean=0&si$

As t statistics falls in the rejection region we reject the null hypothesis

Hence we have sufficient evidence to support the claim that mean IQ score, $\mu$, f this year's class is greater than that of previous years.

So we can conclude, using the 0.05 level of significance, mean IQ score, $\mu$, f this year's class is greater than that of previous years. Yes.