91Ó°ÊÓ

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

We need to test

$H_0:\mathrm{μ}=40$versus $H_a:\mathrm{μ}>40$using $\mathrm{Î\pm }=0.05$.

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 $\mathrm{μ}=40$yards, with a standard deviation of $\mathrm{σ}=2$yards.

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

Distribution for the test is normal i.e.

$\stackrel{¯}{X}:\mathcal{N}\left(\mathrm{μ},\frac{\mathrm{σ}}{\sqrt{n}}\right)=\mathcal{N}\left(40,\frac{2}{\sqrt{20}}\right)$

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

P- value $=P(\stackrel{¯}{x}>45)=1-P(\stackrel{¯}{x}<45)≈0$

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

We know that,

$\mathrm{Î\pm }=0.05>0=P$value

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

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.