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Chapter 12: Problem 1

From the given information in each case below, state what you know about the \(P\) -value for a chi-square test and give the conclusion for a significance level of \(\alpha=.01\). a. \(X^{2}=7.5, \mathrm{df}=2\) b. \(X^{2}=13.0, \mathrm{df}=6\) c. \(\quad X^{2}=18.0, \mathrm{df}=9\) d. \(\quad X^{2}=21.3, \mathrm{df}=4\) e. \(\quad X^{2}=5.0, \mathrm{df}=3\)

Short Answer

Expert verified
We need a statistical calculator or software to calculate the exact P-value based on the provided \(X^{2}\) and df values. Without these, we can't provide an exact answer. But generally, we would compare the calculated P-value with the significance level, and if it's less, we would reject the null hypothesis, meaning the result is statistically significant. Otherwise, we couldn't reject the null hypothesis, implying the result is not statistically significant.

Step by step solution

01

Understanding chi-square and P-value

The chi-square (\(X^{2}\)) is a statistical measure used in the context of a chi-square test. The degrees of freedom (df) generally refers to the number of values in the final calculation that are free to vary. The P-value is the probability that, assuming the null hypothesis is true, the statistical summary would be greater than or equal to the actual observed results.
02

Calculate P-value

Based on the provided chi-square and degrees of freedom for each case, use the chi-square distribution table to find the P-value. This could also be achieved by using any statistical software or calculator.
03

Comparing P-value with significance level

The final step is to compare the calculated P-value with your chosen significance level (\(\alpha = .01\)). If the P-value is less than or equal to the significance level, reject the null hypothesis, meaning that the results are statistically significant. If the P-value is greater than \(\alpha\), there is not enough evidence to reject the null hypothesis, meaning the results aren鈥檛 statistically significant.
04

Deriving the conclusions

Finally, it's important to write out the conclusions for each of the cases in the exercise in simple terms, where rejecting the null hypothesis means there is a statistically significant result and not rejecting the null hypothesis means there is no statistically significant result.

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Key Concepts

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

P-value
The P-value is a crucial concept in statistical hypothesis testing, especially for the chi-square test. It helps decide whether the observed results are significant under the null hypothesis. Think of the P-value as a measure of surprise. How shocked should you be by your data if the null hypothesis is actually true?
  • If the P-value is very small, it suggests that the data we have observed would be highly unlikely under the null hypothesis.
  • An extremely small P-value typically means we have statistical evidence against the null hypothesis.
P-values are compared against a pre-determined significance level (like 0.01) to decide the outcome of our test. Use the chi-square distribution table or a calculator to find the P-value for a given chi-square statistic and degrees of freedom. This provides an easy way to figure out whether you should reject or not reject the null hypothesis.
Degrees of Freedom
Degrees of freedom (often abbreviated as df) are an important consideration in the chi-square test. They tell us how many values in our calculation are free to vary and play a key role in finding the P-value.
  • Typically, in a chi-square test, the degrees of freedom are calculated as the number of possible categories minus 1.
  • For instance, if you have 5 categories, your degrees of freedom would be 4.
The concept of degrees of freedom is a little tricky because it depends on the context of the problem. But once known, it's easy to match the chi-square statistic with its corresponding P-value using a chi-square distribution table. This match helps in forming conclusions about the significance of your test.
Significance Level
The significance level, often denoted by the Greek letter alpha (\(\alpha\)), is a threshold set by the researcher to decide whether to reject the null hypothesis. Common choices for \(\alpha\) are 0.05, 0.01, or 0.10, with smaller values implying stricter criteria for significance.
  • If the P-value is less than or equal to \(\alpha\), the results are significant, and the null hypothesis is rejected.
  • If the P-value is greater than \(\alpha\), we do not have enough evidence to reject the null hypothesis.
By setting \(\alpha\) to 0.01, as in our exercise, we are requiring stronger evidence to claim statistical significance. This means the risk of wrongly rejecting a true null hypothesis (Type I error) is reduced. Significance level acts as a gatekeeper to decide on retaining or rejecting the null hypothesis.
Null Hypothesis
The null hypothesis is a statement of no effect or no difference, often symbolized as \(H_0\). In the context of the chi-square test, it proposes that any observed difference in data is purely due to chance.
  • The null hypothesis is usually the statement being tested, and the goal is to determine whether there is enough statistical evidence to reject it.
  • If you reject the null hypothesis, it suggests there is a statistically significant effect or difference.
  • Failing to reject it means the observed data is consistent with what we would expect under the null hypothesis.
It is important to understand that in hypothesis testing, failing to reject the null hypothesis is not the same as proving it true. Instead, it indicates insufficient evidence to support an alternative hypothesis. The chi-square test provides a framework to evaluate the null hypothesis using statistics like P-value, degrees of freedom, and significance level.

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Most popular questions from this chapter

The authors of the article "A Survey of Parent Attitudes and Practices Regarding Underage Drinking" (Journal of youth and Adolescence [1995]: 315-334) conducted a telephone survey of parents with preteen and teenage children. One of the questions asked was "How effective do you think you are in talking to your children about drinking?" Responses are summarized in the accompanying \(3 \times 2\) table. Using a significance level of \(.05\), carry out a test to determine whether there is an association between age of children and parental response. $$ \begin{array}{l|cc} & \text { Age of Children } \\ \hline \text { Response } & \text { Preteen } & \text { Teen } \\ \hline \text { Very Effective } & 126 & 149 \\ \text { Somewhat Effective } & 44 & 41 \\ \text { Not at All Effective or Don't Know } & 51 & 26 \\ \hline \end{array} $$

A particular state university system has six campuses. On each campus, a random sample of students will be selected, and each student will be categorized with respect to political philosophy as liberal, moderate, or conservative. The null hypothesis of interest is that the proportion of students falling in these three categories is the same at all six campuses. a. On how many degrees of freedom will the resulting \(X^{2}\) test be based? b. How does your answer in Part (a) change if there are seven campuses rather than six? c. How does your answer in Part (a) change if there are four rather than three categories for political philosophy?

The article 鈥淩egional Differences in Attitudes Toward Corporal Punishment" (Journal of Marriage and Family [1994]: 314-324) presents data resulting from a random sample of 978 adults. Each individual in the sample was asked whether he or she agreed with the following statement: "Sometimes it is necessary to discipline a child with a good, hard spanking." Respondents were also classified according to the region of the United States in which they lived. The resulting data are summarized in the accompanying table. Is there an association between response (agree, disagree) and region of residence? Use \(\alpha=.01\). $$ \begin{array}{l|cc} & \text { Response } \\ \hline \text { Region } & \text { Agree } & \text { Disagree } \\ \hline \text { Northeast } & 130 & 59 \\ \text { West } & 146 & 42 \\ \text { Midwest } & 211 & 52 \\ \text { South } & 291 & 47 \\ \hline \end{array} $$

The paper 鈥淥verweight Among Low-Income Preschool Children Associated with the Consumption of Sweet Drinks" (Pediatrics [2005]: 223-229) described a study of children who were underweight or normal weight at age 2. Children in the sample were classified according to the number of sweet drinks consumed per day and whether or not the child was overweight one year after the study began. Is there evidence of an association between whether or not children are overweight after one year and the number of sweet drinks consumed? Assume that it is reasonable to regard the sample of children in this study as representative of 2 - to 3 -year-old children and then test the appropriate hypotheses using a \(.05\) significance level. $$ \begin{array}{c|rc} \text { Number of Sweet } & \text { Overweight? } \\ \begin{array}{c} \text { Drinks Consumed } \\ \text { per Day } \end{array} & \text { Yes } & \text { No } \\ \hline 0 & 22 & 930 \\ 1 & 73 & 2074 \\ 2 & 56 & 1681 \\ 3 \text { or More } & 102 & 3390 \\ \hline \end{array} $$

In November 2005 , an international study to assess public opinion on the treatment of suspected terrorists was conducted ("Most in U.S., Britain, S. Korea and France Say Torture Is OK in at Least Rare Instances," Associated Press, December 7,2005 ). Each individual in random samples of 1000 adults from each of nine different countries was asked the following question: "Do you feel the use of torture against suspected terrorists to obtain information about terrorism activities is justified?" Responses consistent with percentages given in the article for the samples from Italy, Spain, France, the United States, and South Korea are summarized in the table at the top of the next page. Based on these data, is it reasonable to conclude that the response proportions are not the same for all five countries? Use a .01 significance level to test the appropriate hypotheses. $$ \begin{array}{l|ccccc} & & & \text { Response } \\ \hline \text { Country } & \text { Never } & \text { Rarely } & \begin{array}{c} \text { Some- } \\ \text { times } \end{array} & \text { Often } & \begin{array}{c} \text { Not } \\ \text { Sure } \end{array} \\ \hline \text { Italy } & 600 & 140 & 140 & 90 & 30 \\ \text { Spain } & 540 & 160 & 140 & 70 & 90 \\ \text { France } & 400 & 250 & 200 & 120 & 30 \\ \text { United } & 360 & 230 & 270 & 110 & 30 \\ \begin{array}{c} \text { States } \\ \text { South } \\ \text { Korea } \end{array} & 100 & 330 & 470 & 60 & 40 \\ \hline \end{array} $$
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