Chapter 10, Problem 10.61RE

To determine

To Explain: that these data provide good evidence that response times have decreased since last year and carry out a significance test to find out.

Answer to Problem 10.61RE

There is enough evidence to help the claim that response times have decreased since last year.

Explanation of Solution

Given:

  $\begin{array}{c}\overline{x}=6.48\\ s=2\\ n=400\end{array}$

Claim: mean is smaller than 6.7 (decreased)

Formula used:

  $z=\frac{\overline{x}-{\mu }_0}{\sigma /\sqrt{n}}$

Calculation:

Assume

  $\alpha =0.05$

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is the population mean is equal to the value given in claim. If the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $\begin{array}{c}{H}_0:\mu =67\\ H_1:\mu <6.7\end{array}$

The sample size of $n=400$ is large, therefore it can use the normal distribution. The sampling distribution of the sample mean has mean $\mu$ and standard deviation $\frac{\sigma }{\sqrt{n}}\cdot$

The z-value is

  $\begin{array}{c}z=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}\\ =\frac{6.48-6.7}{2/\sqrt{400}}\\ =-2.20\end{array}$

The P-value is the probability of getting the value more extreme or equal to the standardized test statistic $z$ , assuming that the null hypothesis is true.

  $P=P\left(Z<-2.20\right)=0.0139$

If the P-value is lesser than the significance level $\alpha$ , then the null hypothesis is rejected.

  $P<0.05\Rightarrow {\text{Reject H}}_0$

There is enough evidence to help the claim that response times have decreased since last year.