Question

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and, or critical value, and state the conclusion.

Kentucky Derby The table below lists the frequency of wins for different post positions through the 141st running of the Kentucky Derby horse race. A post position of 1 is closest to the inside rail, so the horse in that position has the shortest distance to run. (Because the number of horses varies from year to year, only the first 10 post positions are included.) Use a 0.05 significance level to test the claim that the likelihood of winning is the same for the different post positions. Based on the result, should bettors consider the post position of a horse racing in the Kentucky Derby?

Post Position	1	2	3	4	5	6	7	8	9	10
Wins	19	14	11	15	15	7	8	12	5	11

Short answer

There is not enough evidence to conclude that the probability of winning is not the same for different post positions.

Since there is no significant difference in the frequency of wins for the different post positions, there isn’t a need for the bettors to consider the post position of a horse in the Kentucky Derby.

Step-by-step solution

Given information

The frequency of wins is provided for the different post positions in the 141 Kentucky Derby horse races.

Check the requirements of the test

Assume the experimental units are obtained by random sampling.

If the expected frequencies are larger than 5, the requirements of the test are fulfilled.

Let O denote the observed frequencies of the wins.

The following values are obtained:

\(\begin{aligned}{l}{O_1} = 19\\{O_2} = 14\\{O_3} = 11\\{O_4} = 15\end{aligned}\)

\(\begin{aligned}{l}{O_5} = 15\\{O_6} = 7\\{O_7} = 8\\{O_8} = 12\end{aligned}\)

\(\begin{aligned}{l}{O_9} = 5\\{O_{10}} = 11\end{aligned}\)

The sum of all observed frequencies is computed below:

\(\begin{aligned}{c}n = 19 + 14 + .... + 11\\ = 117\end{aligned}\)

Let E denote the expected frequencies.It is given that the number of wins is expected to be the same for all the positions.

The expected frequencies for each of the 10 positions are equal to:

\(\begin{aligned}{c}E = \frac{{117}}{{10}}\\ = 11.7\end{aligned}\)

Thus, the requirements of the test are satisfied.

State the hypotheses

The null hypothesis for conducting the given test is as follows:

\({H_0}:\)The probability of winning is the same for different post positions.

The alternative hypothesis is as follows:

\({H_a}:\)The probability of winning is not the same for different post positions.

Conduct the hypothesis test

The table below shows the necessary calculations:

Post Position	O	E	\(\left( {O - E} \right)\)	\({\left( {O - E} \right)^2}\)	\(\frac{{{{\left( {O - E} \right)}^2}}}{E}\)
1	19	11.7	7.3	53.29	4.5547
2	14	11.7	2.3	5.29	0.4521
3	11	11.7	-0.7	0.49	0.0419
4	15	11.7	3.3	10.89	0.9308
5	15	11.7	3.3	10.89	0.9308
6	7	11.7	-4.7	22.09	1.8880
7	8	11.7	-3.7	13.69	1.1701
8	12	11.7	0.3	0.09	0.0077
9	5	11.7	-6.7	44.89	3.8367
10	11	11.7	-0.7	0.49	0.0419

The value of the test statistic is equal to:

\(\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \;\;\\ = 4.5547 + 0.4521 + ... + 0.0419\\ = 13.855\end{aligned}\)

Thus,\({\chi ^2} = 13.855\).

Let k be the number of posts that are equal to 10.

The degrees of freedom for\({\chi ^2}\)is computed below:

\(\begin{aligned}{c}df = k - 1\\ = 10 - 1\\ = 9\end{aligned}\)

State the decision

The chi-square table is used to determine the critical value of\({\chi ^2}\)at\(\alpha = 0.05\)with 9 degrees of freedom as 16.919.

The p-value is equal to,

\(\begin{aligned}{c}p - value = P\left( {{\chi ^2} > 13.855} \right)\\ = 0.128\end{aligned}\)

Since the test statistic value is less than the critical value and the p-value is greater than 0.05, the null hypothesis is failed to be rejected.

Determine the conclusion

There is enough evidence to conclude that the probability of winning is same for different post positions.

Since there is no significant difference in the frequency of wins for the different post positions, there isn’t the need for the bettors to consider the post position of a horse in the Kentucky Derby.