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Chapter 12: Problem 15

A random sample of 1000 registered voters in a certain county is selected, and each voter is categorized with respect to both educational level (four categories) and preferred candidate in an upcoming election for county supervisor (five possibilities). The hypothesis of interest is that educational level and preferred candidate are independent. a. If \(X^{2}=7.2\), what would you conclude at significance level. \(10 ?\) b. If there were only four candidates running for election, what would you conclude if \(X^{2}=14.5\) and \(\alpha=.05 ?\)

Short Answer

Expert verified
For part a, if the \(X^{2}\) value of 7.2 is less than the critical value \(C_{1}\), which must be identified, then the preferred candidate is independent of educational level at significance level 0.10. For part b, if the \(X^{2}\) value of 14.5 is greater than the critical value \(C_{2}\) at .05 significance level, then educational level is not independent of preferred candidate. The actual conclusion depends on the comparison between the calculated \(X^{2}\) and critical values \(C_{1}, C_{2}\), which can not be provided in this context due to lack of a chi-square distribution table or statistical software.

Step by step solution

01

Identify degrees of freedom, and obtain critical value

For part a, we have educational level (4 categories) and preferred candidate (5 possibilities), so there are (4-1) * (5-1) = 12 degrees of freedom. The critical value at significance level .10 (or 10%) and 12 degrees of freedom can be obtained from the chi-square distribution table or by using statistical software. Let say the critical value from table is \(C_{1}\).
02

Compare \(X^{2}\) value to critical value

If the \(X^{2}\) value of 7.2 is less than the critical value \(C_{1}\), then we fail to reject the null hypothesis of independence. If the \(X^{2}\) value is greater than the critical value, then we reject the null hypothesis and conclude that educational level and preferred candidate are related.
03

Repeat steps for part b

For part b, assuming there are still 4 categories for education, the degrees of freedom due to 4 candidates running for election would be (4-1)*(4-1) = 9. Calculate the critical value at significance level of .05 (or 5%) and 9 degrees of freedom. Let's denote this critical value as \(C_{2}\). Then the \(X^{2}\) value of 14.5 needs to be compared to this critical value \(C_{2}\).
04

Final interpretation

Just like in Step 2, if the \(X^{2}\) value is less than the critical value \(C_{2}\), we fail to reject the null hypothesis. If it's greater, we reject the null hypothesis and conclude that educational level is not independent of preferred candidate.

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Key Concepts

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

Degrees of Freedom
When dealing with the Chi-Square Test of Independence, understanding degrees of freedom is crucial. Degrees of freedom refer to the number of values or categories that are free to vary when calculating statistical measures. In a Chi-Square test, the degrees of freedom are determined by the number of categories in each variable you are examining.

To calculate the degrees of freedom in our exercise, we need to know the number of categories for both variables. For part (a), the educational level has 4 categories, and the preferred candidate has 5 possibilities. Therefore, the degrees of freedom are calculated as follows:
  • Degrees of Freedom (df) = (Number of rows - 1) * (Number of columns - 1)
  • In this case, df = (4 - 1) * (5 - 1) = 12
For part (b), the number of preferred candidates changes to 4, which recalculates the degrees of freedom to:
  • df = (4 - 1) * (4 - 1) = 9
Calculating the correct degrees of freedom is essential as it influences the critical value you will use to determine the test's outcome.
Critical Value
The critical value in a Chi-Square Test of Independence is a threshold that determines whether you should reject the null hypothesis. It is obtained from a Chi-Square distribution table or statistical software based on your significance level and degrees of freedom.

For part (a) of our exercise, we have a significance level of 0.10 and 12 degrees of freedom. The critical value, denoted as \(C_1\), is the point beyond which the probability of observing such an extreme value (if the null hypothesis were true) is less than 10%. Similarly, for part (b), with 9 degrees of freedom and a significance level of 0.05, the critical value is denoted as \(C_2\).
  • A Chi-Square value less than the critical value means you fail to reject the null hypothesis.
  • A Chi-Square value greater than the critical value leads to the rejection of the null hypothesis.
Finding and understanding the critical value is crucial to deciding whether or not our observed data supports the independence of the variables.
Null Hypothesis
The null hypothesis in the context of a Chi-Square Test of Independence posits that the two variables in question are independent. In other words, it suggests that there is no relationship between them.

In our exercise scenario, the null hypothesis is that the educational level of voters is independent of their preferred candidate. When conducting the Chi-Square test, our goal is to determine if the data provides enough evidence to reject this hypothesis.
  • If the Chi-Square statistic calculated from the data is less than the critical value, we have insufficient evidence to reject the null hypothesis. We therefore conclude that there is no significant relationship.
  • If the Chi-Square statistic exceeds the critical value, we reject the null hypothesis, suggesting a relationship exists.
This decision-making process relies on the comparison between the Chi-Square statistic and the critical value, highlighting whether or not a statistically significant relationship exists between the variables being tested.

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Most popular questions from this chapter

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.

The article "Cooperative Hunting in Lions: The Role of the Individual" (Behavioral Ecology and Sociobiology [1992]: \(445-454\) ) discusses the different roles taken by lionesses as they attack and capture prey. The authors were interested in the effect of the position in line as stalking occurs; an individual lioness may be in the center of the line or on the wing (end of the line) as they advance toward their prey. In addition to position, the role of the lioness was also considered. A lioness could initiate a chase (be the first one to charge the prey), or she could participate and join the chase after it has been initiated. Data from the article are summarized in the accompanying table. $$ \begin{array}{l|cc} & \text { Role } \\ \hline \text { Position } & \text { Initiate Chase } & \text { Participate in Chase } \\ \hline \text { Center } & 28 & 48 \\ \text { Wing } & 66 & 41 \\ \hline \end{array} $$ Is there evidence of an association between position and role? Test the relevant hypotheses using \(\alpha=.01\). What assumptions about how the data were collected must be true for the chi-square test to be an appropriate way to analyze these data?

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 \(\alpha=.05\). $$ \begin{array}{lccc} & & \text { Character Type } \\ \hline \text { Sex } & \text { Positive } & \text { Negative } & \text { Neutral } \\ \hline \text { Male } & 255 & 106 & 130 \\ \text { Female } & 85 & 12 & 49 \\ \hline \end{array} $$

Each boy in a sample of Mexican American males, age 10 to 18 , was classified according to smoking status and response to a question asking whether he likes to do risky things. The following table is based on data given in the article "The Association Between Smoking and Unhealthy Behaviors Among a National Sample of Mexican-American Adolescents" (Journal of School Health [1998]: 376-379): $$ \begin{array}{l|cr} & \text { Smoking Status } \\ \hline & \text { Smoker } & \text { Nonsmoker } \\ \hline \text { Likes Risky Things } & 45 & 46 \\ \text { Doesn't Like Risky Things } & 36 & 153 \\ \hline \end{array} $$ Assume that it is reasonable to regard the sample as a random sample of Mexican-American male adolescents. a. Is there sufficient evidence to conclude that there is an association between smoking status and desire to do risky things? Test the relevant hypotheses using \(\alpha=.05 .\) b. Based on your conclusion in Part (a), is it reasonable to conclude that smoking causes an increase in the desire to do risky things? Explain.

Using data from a national survey, the authors of the paper "What Do Happy People Do?" (Social Indicators Research [2008]: 565-571) concluded that there was convincing evidence of an association between amount of time spent watching television and whether or not a person reported that they were happy. They observed that unhappy people tended to watch more television. The authors write: This could lead us to two possible interpretations: 1\. Television viewing is a pleasurable enough activity with no lasting benefit, and it pushes aside time spent in other activities - ones that might be less immediately pleasurable, but that would provide long-term benefits in one's condition. In other words, television does cause people to be less happy. 2\. Television is a refuge for people who are already unhappy. TV is not judgmental nor difficult, so people with few social skills or resources for other activities can engage in it. Furthermore, chronic unhappiness can be socially and personally debilitating and can interfere with work and most social and personal activities, but even the unhappiest people can click a remote and be passively entertained by a TV. In other words, the causal order is reversed for people who watch television; unhappiness leads to television viewing. Using only data from this study, do you think it is possible to determine which of these two conclusions is correct? If so, which conclusion do you think is correct and why? If not, explain why it is not possible to decide which conclusion is correct based on the study data.
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