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Chapter 12: Problem 2

A particular paperback book is published in a choice of four different covers. A certain bookstore keeps copies of each cover on its racks. To test the hypothesis that sales are equally divided among the four choices, a random sample of 100 purchases is identified. a. If the resulting \(X^{2}\) value were \(6.4\), what conclusion would you reach when using a test with significance level .05? b. What conclusion would be appropriate at significance level \(.01\) if \(X^{2}=15.3\) ? c. If there were six different covers rather than just four, what would you conclude if \(X^{2}=13.7\) and a test with \(\alpha=.05\) was used?

Short Answer

Expert verified
a) We would conclude that the sales are equally divided among the four covers. \n b) We would conclude that the sales aren't equally divided among the four covers.\n c) We would conclude that the sales aren't equally divided among the six covers.

Step by step solution

01

Understanding the significance level and degrees of freedom

This step involves first understanding the concept of the significance level. The significance level, often denoted as alpha or \(\alpha\), is a threshold below which the null hypothesis may be rejected. Usually, if the calculated statistic is less than the significance level, the null hypothesis is accepted. Otherwise, it is rejected. In this case, the null hypothesis in all three scenarios is that the distribution of cover preferences is uniform i.e., the four or six covers are chosen equally by the customers. \n\nNext, we define the degrees of freedom. In this context, the degrees of freedom are defined as \(df = C - 1\), where \(C\) is the number of different covers. This means there are 3 degrees of freedom for scenarios a and b, and 5 degrees of freedom for scenario c.
02

Comparing the obtained \(\chi^{2}\) value with the critical \(\chi^{2}\) value

To conduct the test, we need to compare the obtained \(\chi^{2}\) value from the random sample with the critical \(\chi^{2}\) value at a given level of significance (alpha = .05 in scenarios a and c, and alpha = .01 in scenario b) and appropriate degrees of freedom (3 in scenarios a and b, and 5 in scenario c).\n\n For the Chi-Square distribution, the critical \(\chi^{2}\) value is the value that the obtained \(\chi^{2}\) value must exceed in order for the null hypothesis to be rejected. Using Chi-Square tables or statistical software, we can find that at 3 degrees of freedom, the critical \(\chi^{2}\) values are roughly 7.815 for \(\alpha=.05\) and 11.345 for \(\alpha=.01\). At 5 degrees of freedom, the critical \(\chi^{2}\) value is about 11.070 for \(\alpha=.05\).\n\nTo summarize, for each scenario we compare the obtained \(\chi^{2}\) value with the respective critical \(\chi^{2}\) value.
03

Making conclusions

In step 3, we make conclusions based on our analysis. For scenario a, since the observed \(\chi^{2}\) value (6.4) is less than the critical \(\chi^{2}\) value at 3 degrees of freedom for \(\alpha=.05\) (7.815), we would accept the null hypothesis. That suggests that the sales are equally divided among the four covers.\n\nFor scenario b, the observed \(\chi^{2}\) value (15.3) is greater than the critical \(\chi^{2}\) value at 3 degrees of freedom for \(\alpha=.01\) (11.345). So, we would reject the null hypothesis. This suggests that the sales aren't equally divided among the four covers.\n\nFinally, for scenario c, our observed \(\chi^{2}\) value (13.7) exceeds the critical \(\chi^{2}\) value for \(\alpha=.05\) at 5 degrees of freedom (11.070). So, in this case, we would also reject the null hypothesis, suggesting that the sales aren't equally divided among the six covers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significance Level
In statistical hypothesis testing, the significance level, denoted as \( \alpha \), is a critical probability threshold that determines when we can reject the null hypothesis. Essentially, it indicates the likelihood of the study rejecting the null hypothesis, given that it were true.

Typically, a significance level of \(0.05\) or \(5%\) is used in most research, suggesting a 5% risk of concluding that a difference exists when there is no actual difference. A lower \( \alpha \) such as \(0.01\) means a stricter threshold and less tolerance for error, reducing the chance of a Type I error (incorrectly rejecting a true null hypothesis).

In the context of the exercise, for a result to be deemed statistically significant at the \( \alpha = 0.05\) level, the observed Chi-Square \( X^2 \) value must exceed the critical value associated with that level. If it does not, we fail to reject the null hypothesis, suggesting that the observed data fit the expected distribution.
Degrees of Freedom
Degrees of freedom, often abbreviated as \(df\), is a parameter that limits the amount of independent information in the data being analyzed. In the Chi-Square test, the degrees of freedom depend on the number of categories being analyzed.

Specifically, for a test like the one described in the exercise, the degrees of freedom are calculated by taking the number of categories (\(C\)) and subtracting 1, thus \(df = C - 1\). The degrees of freedom influence the shape and scale of the Chi-Square distribution, which is used to determine the critical value.

In our textbook example, scenarios a and b have 3 degrees of freedom (\(C=4\), \(df=4-1\)), while scenario c has 5 degrees of freedom (\(C=6\), \(df=6-1\)). The choice of degrees of freedom is essential for finding the correct critical value and thus for making the correct inference about the null hypothesis.
Null Hypothesis
The null hypothesis, commonly represented as \(H_0\), is a default statement that there is no effect or no difference, which the test seeks to either reject or fail to reject based on the sample data. It is the assumption that any kind of difference or significance you see in a set of data is due to chance.

In the exercise scenario, the null hypothesis posits that there is no preference among the different book covers and that they are equally likely to be chosen by customers. Statistical tests, including Chi-Square tests, are used to analyze sample data to determine whether the null hypothesis can be accepted or rejected. If the observed statistic is less than the critical value for the given degrees of freedom and significance level, the null hypothesis is not rejected. If it exceeds the critical value, then the null hypothesis can be rejected in favor of an alternative hypothesis.
Critical Value
The critical value in a Chi-Square test is the cutoff point that dictates whether the null hypothesis should be rejected. This value is derived from the Chi-Square distribution for a specific level of significance (\( \alpha \) level) and the degrees of freedom associated with the test.

To decide on the null hypothesis, the critical value is compared against the calculated Chi-Square statistic from the data. If the calculated value is greater than the critical value, the null hypothesis is rejected; if it's less, the null hypothesis cannot be rejected.

In the exercise scenarios, critical values are determined for the significance levels of \(0.05\) and \(0.01\) with respective degrees of freedom. In scenario a, for \( \alpha = 0.05\) and \(df=3\), the result \(X^2 = 6.4\) does not exceed the critical value, leading us to accept the null hypothesis. However, in scenarios b and c, the \(X^2\) values do exceed their respective critical values at given significance levels, prompting us to reject the null hypothesis. Knowing how to find and interpret critical values is pivotal for correctly conducting hypothesis tests.

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Most popular questions from this chapter

The authors of the paper "Movie Character Smoking and Adolescent Smoking: Who Matters More, Good Guys or Bad Guys?" (Pediatrics [2009]: 135-141) classified characters who were depicted smoking in movies released between 2000 and \(2005 .\) The smoking characters were classified according to sex and whether the character type was positive, negative or neutral. The resulting data is given in the accompanying table. Assume that it is reasonable to consider this sample of smoking movie characters as representative of smoking movie characters. Do the data provide evidence of an association between sex and character type for movie characters who smoke? Use \(\alpha=.05\). $$ \begin{array}{lccc} & & \text { Character Type } \\ \hline \text { Sex } & \text { Positive } & \text { Negative } & \text { Neutral } \\ \hline \text { Male } & 255 & 106 & 130 \\ \text { Female } & 85 & 12 & 49 \\ \hline \end{array} $$

The authors of the paper 鈥淩acial Stereotypes in Children's Television Commercials鈥� (Journal of Advertising Research [2008]: 80-93) counted the number of times that characters of different ethnicities appeared in commercials aired on Philadelphia television stations, resulting in the data in the accompanying table. $$ \begin{array}{l|cccc} \hline \text { Ethnicity } & \begin{array}{l} \text { African- } \\ \text { American } \end{array} & \text { Asian } & \text { Caucasian } & \text { Hispanic } \\ \hline \begin{array}{c} \text { Observed } \\ \text { Frequency } \end{array} & 57 & 11 & 330 & 6 \\ \hline \end{array} $$ Based on the 2000 Census, the proportion of the U.S. population falling into each of these four ethnic groups are \(.177\) for African-American, \(.032\) for Asian, \(.734\) for Caucasian, and \(.057\) for Hispanic. Do the data provide sufficient evidence to conclude that the proportions appearing in commercials are not the same as the census proportions? Test the relevant hypotheses using a significance level of .01.

The report "Fatality Facts 2004: Bicydes" (Insurance Institute, 2004) included the following table classifying 715 fatal bicycle accidents according to time of day the accident occurred. $$ \begin{array}{lc} \text { Time of Day } & \text { Number of Accidents } \\ \hline \text { Midnight to } 3 \text { A.M. } & 38 \\ \text { 3 A.M. to } 6 \text { A.M. } & 29 \\ \text { 6 A.M. to } 9 \text { A.M. } & 66 \\ \text { 9A.M. to Noon } & 77 \\ \text { Noon to } 3 \text { P.M. } & 99 \\ \text { 3 P.M. to } 6 \text { r.M. } & 127 \\ \text { 6 P.M. to 9 p.M. } & 166 \\ \text { 9 P.M. to Midnight } & 113 \\ \hline \end{array} $$ a. Assume it is reasonable to regard the 715 bicycle accidents summarized in the table as a random sample of fatal bicycle accidents in 2004 . Do these data support the hypothesis that fatal bicycle accidents are not equally likely to occur in each of the 3 -hour time periods used to construct the table? Test the relevant hypotheses using a significance level of \(.05\). b. Suppose a safety office proposes that bicycle fatalities are twice as likely to occur between noon and midnight as during midnight to noon and suggests the following hypothesis: \(H_{0}: p_{1}=1 / 3, p_{2}=2 / 3\), where \(p_{1}\) is the proportion of accidents occurring between midnight and noon and \(p_{2}\) is the proportion occurring between noon and midnight. Do the given data provide evidence against this hypothesis, or are the data consistent with it? Justify your answer with an appropriate test. (Hint: Use the data to construct a one-way table with just two time categories.)

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Does including a gift with a request for a donation affect the proportion who will make a donation? This question was investigated in a study described in the report 鈥淕ift-Exchange in the Field" (Institute for the Study of Labor, 2007). In this study, letters were sent to a large number of potential donors in Germany. The letter requested a donation for funding schools in Bangladesh. Those who were to receive the letter were assigned at random to one of three groups. Those in the first group received the letter with no gift. Those in the second group received a letter that included a small gift (a postcard), and those in the third group received a letter with a larger gift (four postcards). The response of interest was whether or not a letter resulted in a donation. $$ \begin{array}{l|cc} & \text { Donation } & \text { No Donation } \\ \hline \text { No Gift } & 397 & 2865 \\ \text { Small Gift } & 465 & 2772 \\ \text { Large Gift } & 691 & 2656 \\ \hline \end{array} $$ a. Carry out a hypothesis test to determine if there is convincing evidence that the proportions in the two donation categories are not the same for all three types of requests. Use a significance level of \(.01\). b. Based on your analysis in Part (a) and a comparison of observed and expected cell counts, write a brief description of how the proportion making a donation varies for the three types of request.
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