Question

The mean throwing distance of a football for Marco, a high school freshman quarterback, is $40$yards, with a standard deviation of two yards. The team coach tells Marco to adjust his grip to get more distance. The coach records the distances for $20$throws. For the $20$throws, Marco’s mean distance was $45$yards. The coach thought the different grip helped Marco throw farther than $40$yards. Conduct a hypothesis test using a preset $\alpha =0.05$. Assume the throw distances for footballs are normal.

First, determine what type of test this is, set up the hypothesis test, find the$p$-value, sketch the graph, and state your conclusion.

Short answer

We reject $H_0:\mu =40$.

We don't think the average pitching distance of Marco's soccer is in $40$yards but adjusting the grip will increase the distance.

Step-by-step solution

Given Information

This is a test of a single population mean, because the problem is about Marco's mean distance.

We need to test

$H_0:\mu =40$versus $H_a:\mu >40$using $\alpha =0.05$.

For Mark's further throw, his median distance will be greater than $40$yards. The $>$tells as this is right-tailed.

The mean throwing distance of a football for Marco, a high school freshman quarterback, is $\mu =40$yards, with a standard deviation of $\sigma =2$yards.

Explanation

Random variable $\overline{X}$is the mean throwing distance of a football.

Distribution for the test is normal i.e.

$\overline{X}:\mathcal{N}\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)=\mathcal{N}\left(40,\frac{2}{\sqrt{20}}\right)$

Now we calculate the p-value using the normal distribution for a mean

P- value $=P(\overline{x}>45)=1-P(\overline{x}<45)\approx 0$

Where the sample mean in the problem is given as $45$

The $p-$value denotes the probability to the right of the sample mean in the normal distribution

Conclusion

We know that,

$\alpha =0.05>0=P$value

We reject ${\mathrm{H}}_0:\mathrm{\mu }=40$

In other words, we do not think the mean throwing distance of a football for Marco is $40$yards but more distance when adjust his grip.

At thelocalid="1650297247231" $5\%$significance level, we conclude that Marco get more distance when adjust hos grip.
The sample data show there is sufficient evidence that Marco's mean distance is greater than localid="1650297250105" $40$yards.