8.10 Examples  (Page 2/6)

Set up the Hypothesis Test:

Since the problem is about a mean weight, this is a test of a single population mean .

$H_{\mathrm{o}}$ : $\mu$ $=275$ $H_a$ : $\mu$ $>275$ This is a right-tailed test.

Calculating the distribution needed:

Random variable: $\overline{X}$ = the mean weight, in pounds, lifted by the football players.

Distribution for the test: It is normal because $\sigma$ is known.

$\overline{X}$ ~ $N$ $(275,\frac{55}{\sqrt{30}})$

$\overline{x}$ $=286.2$ pounds (from the data).

$\sigma =55$ pounds (Always use $\sigma$ if you know it.) We assume $\mu =275$ pounds unless our data shows us otherwise.

Calculate the p-value using the normal distribution for a mean and using the sample mean as input (see the calculator instructions below for using the data as input):

$\text{p-value}=P($ $\overline{x}$ $>286.2$ ) $=0.1323$ .

Interpretation of the p-value: If $H_{\mathrm{o}}$ is true, then there is a 0.1331 probability (13.23%) that the football players can lift a mean weight of 286.2pounds or more. Because a 13.23% chance is large enough, a mean weight lift of 286.2 pounds or more is not a rare event.

Compare $\alpha$ and the p-value:

$\alpha =0.025\text{p-value}=0.1323$

Make a decision: Since $\alpha$ $$ $\text{p-value}$ , do not reject $H_{\mathrm{o}}$ .

Conclusion: At the 2.5% level of significance, from the sample data, there is not sufficient evidence to conclude that the true mean weight lifted is more than 275pounds.

The p-value can easily be calculated using the TI-83+ and the TI-84 calculators:

Put the data and frequencies into lists. Press STAT and arrow over to TESTS . Press 1:Z-Test . Arrow over to Data and press ENTER . Arrow down and enter 275 for ${\mu }_0$ , 55 for $\sigma$ , the name of the list where you put the data, and the name of the list where you put thefrequencies. Arrow down to $\mu :$ and arrow over to $>{\mu }_0$ . Press ENTER . Arrow down to Calculate and press ENTER . The calculator not only calculates the p-value ( $p=0.1331$ , a little different from the above calculation - in it we used the sample mean rounded to one decimal placeinstead of the data) but it also calculates the test statistic (z-score) for the sample mean, the sample mean, and the sample standard deviation. $\mu >275$ is the alternate hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate ). Press ENTER . A shaded graph appears with $z=1.112$ (test statistic) and $p=0.1331$ (p-value). Make sure when you use Draw that no other equations are highlighted in $Y=$ and the plots are turned off.