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Chapter 7: Q 131SE (page 443)

Question: Point spreads of NFL games. Refer to the Chance (Fall 1998) study of point-spread errors in NFL games, Exercise 7.41 (p. 411). Recall that the difference between the actual game outcome and the point spread established by odds makers鈥攖he point-spread error鈥攚as calculated for 240 NFL games. The results are summarized as follows: . Suppose the researcher wants to know whether the true standard deviation of the point spread errors exceeds 15. Conduct the analysis using 伪 = 0.10.

Short Answer

Expert verified

The true standard deviation of the points spread errors is 15.

Step by step solution

01

Given information

The given problem explains the point-spread error in NFL games. The point-spread error was calculated for 240 NFL games.

From the sample, the sample mean is -1.6, and the sample standard deviation is 13.3.

Here, the test is based on the true standard deviation of the point-spread error.

02

 Concept of testing of hypothesis

The null hypothesis, which can be further tested with the use of specific statistical tests, is an assumption that must be made before making any choice regarding the true value of the parameters. The choice is then made based on the calculated value of the parameter and critical, which means that if the computed value falls within the critical zone, the null hypothesis is rejected; otherwise, it is accepted.

03

Testing the #FA3273hypothesis

The null and alternative hypotheses are

Null hypothesis:

H0 : 蟽 = 15

The true standard deviation of points spread errors is 15.

Alternative hypothesis:

Ha : 蟽 > 15

The true standard deviation of points spread errors exceeds 15.

The level of significance is 伪 = 0.10

Since the test statistic is given by

From the given data sample size n is 240.

By substituting the values,

From the table of for df = 239 and 伪 = 0.10 the critical value for the test is,

The calculated statistics value is less than the critical value. That is .

So, the null hypothesis is rejected at the given level of significance, and we conclude that the true standard deviation of the points spread errors is 15.

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Most popular questions from this chapter

For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.

a. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.1\)

b. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.05\)

c. \({H_0}:\mu \ge {\mu _0}\) and \({H_a}:\mu < {\mu _0};\alpha = 0.01\)

d. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.05\)

e. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.1\)

f. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.01\)

g. For each rejection region specified in parts a鈥揻, state the probability notation in z and its respective Type I error value.

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If the rejection of the null hypothesis of a particular test would cause your firm to go out of business, would you want ato be small or large? Explain

Jury trial outcomes. Sometimes, the outcome of a jury trial defies the 鈥渃ommon sense鈥� expectations of the general public (e.g., the 1995 O. J. Simpson verdict and the 2011 Casey Anthony verdict). Such a verdict is more acceptable if we understand that the jury trial of an accused murderer is analogous to the statistical hypothesis-testing process. The null hypothesis in a jury trial is that the accused is innocent. (The status-quo hypothesis in the U.S. system of justice is innocence, which is assumed to be true until proven beyond a reasonable doubt.) The alternative hypothesis is guilt, which is accepted only when sufficient evidence exists to establish its truth. If the vote of the jury is unanimous in favor of guilt, the null hypothesis of innocence is rejected, and the court concludes that the accused murderer is guilty. Any vote other than a unanimous one for guilt results in a 鈥渘ot guilty鈥� verdict. The court never accepts the null hypothesis; that is, the court never declares the accused 鈥渋nnocent.鈥� A 鈥渘ot guilty鈥� verdict (as in the Casey Anthony case) implies that the court could not find the defendant guilty beyond a reasonable doubt

a. Define Type I and Type II errors in a murder trial.

b. Which of the two errors is the more serious? Explain.

c. The court does not, in general, know the values of 伪 and 尾 ; but ideally, both should be small. One of these probabilities is assumed to be smaller than the other in a jury trial. Which one, and why?

d. The court system relies on the belief that the value of is made very small by requiring a unanimous vote before guilt is concluded. Explain why this is so.

e. For a jury prejudiced against a guilty verdict as the trial begins, will the value of伪 increase or decrease? Explain.

f. For a jury prejudiced against a guilty verdict as the trial begins, will the value of 尾 increase or decrease? Explain
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