Chapter 20, Problem 22E

To determine

To test an appropriate hypothesis and state your conclusion and be sure the appropriate assumptions and conditions are satisfied before you proceed.

Answer to Problem 22E

It seems that teams do have home team advantage since there is strong evidence that they win more that $50\%$ of games at home.

Explanation of Solution

It is given in the question that the home team won $136\text{ of the 240}$ regular season national football league games. They want to test that is this strong evidence of a home field advantage in professional football. Thus, the null hypothesis states that the rate is equal to $50\%$ . And the alternative hypothesis states that the rate is morethan $50\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.50\\ H_a:p>0.50\end{array}$

Thus, let us check the appropriate conditions and assumptions for inference to proceed further to test hypothesis as:

Random condition: It is satisfied as we assumed that the sample was randomly selected from the population.

Independent condition: It is satisfied as we can assume that the sample is independent.

  $10\%$ condition: It is satisfied as the sample is less than $10\%$ of all population.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=0.50(240)>10\\ n(1-p)=0.50(240)>10\end{array}$

Thus, all the conditions are satisfied. Thus, for this first we have to select $5:1-\text{PropZInt}$ from the STAT TESTS menu. Then, enter the observed count and sample size for the sample. Then we will indicate what kind of test we want: one-tail lower tail, two-tail, or one-tail upper tail. Then, we will specify whether to calculate the result or draw the result. Hence we have,

  $\begin{array}{l}x=136\\ n=240\\ c=0.95=95\%\end{array}$

So, by putting the values above by using the calculator we will get the result as,

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=2.066\\ P-\text{value}=0.019\end{array}$

And as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

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

Thus, we can conclude that it seems that teams do have home team advantage since there is strong evidence that they win more that $50\%$ of games at home.