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Chapter 12: Problem 12

A certain genetic characteristic of a particular plant can appear in one of three forms (phenotypes). A researcher has developed a theory, according to which the hypothesized proportions are \(p_{1}=.25, p_{2}=.50\), and \(p_{3}=.25 .\) A random sample of 200 plants yields \(X^{2}=\) \(4.63 .\) a. Carry out a test of the null hypothesis that the theory is correct, using level of significance \(\alpha=.05\). b. Suppose that a random sample of 300 plants had resulted in the same value of \(X^{2}\). How would your analysis and conclusion differ from those in Part (a)?

Short Answer

Expert verified
a. At the .05 level, the null hypothesis isn't rejected, suggesting the theory proportions are correct. b. With a 300 sample size, the conclusion would remain the same, though, the evidence against the hypothesized proportions will be weaker.

Step by step solution

01

Understanding the \(X^{2}\) Test

The \(X^{2}\) test is used to determine the difference between observed and expected frequencies. Here, the null hypothesis is that the proportions stated in the theory are correct. To test this hypothesis, we will compare the observed and expected frequencies of the plant characteristics.
02

Calculate the Degrees of Freedom

The degrees of freedom in a chi-square test is calculated by subtracting 1 from the total number of categories (phenotypes) observed. In this case, it's \(df=3-1=2\).
03

Obtain the Critical Value

For the chi-square test, we look up the critical value from the chi-square distribution table. With degrees of freedom 2 and level of significance \(.05\), the critical value is 5.99.
04

Decide to Reject or Not Reject the Null Hypothesis

Since the calculated \(X^{2} = 4.63\) is less than the critical value 5.99, we do not reject the null hypothesis. This suggests that the theorized proportions are correct at the .05 level of significance.
05

Re-analysis with Size 300 sample

If a sample of 300 plants results in the same \(X^{2}\) value (4.63), the conclusion would remain the same. However, it may be considered less conclusive, as the larger sample size was expected to lead to a closer match to the theory. But since the \(X^{2}\) test value remains the same, the statistical evidence does not reject the null hypothesis, regardless of the increase in sample size. The strength of evidence against the hypothesized proportions would be weaker with a larger sample size (300) compared to a sample size of 200.

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Key Concepts

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

Null Hypothesis
In the context of a chi-square test, the null hypothesis ( H_0 ) represents a statement or assumption that there is no effect or no difference. Here, the null hypothesis assumes that the genetic characteristic's proportions are accurate as the researcher's theory suggests. The expected proportions for the plant phenotypes are precisely 0.25 for the first phenotype, 0.50 for the second, and 0.25 for the third.
Testing the null hypothesis requires comparing the observed frequency of characteristics in the sample with what we would expect if the proportions hold true. The purpose is to assess whether the data provides strong enough evidence to overturn this assumption. Failure to reject the null hypothesis implies there is not enough statistical evidence to claim that the proportions are incorrect.
Degrees of Freedom
Degrees of freedom ( df ) in the chi-square test relate to the number of values that can vary in an analysis without breaking any constraints. With the chi-square test in this exercise, it is calculated by subtracting 1 from the total number of categories, which in this case are the different phenotypes of the plant.
  • Since there are 3 phenotypes, the degrees of freedom is calculated as: df = 3 - 1 = 2 .
The degrees of freedom are critical because they influence the shape of the chi-square distribution used to determine the critical value. The more degrees of freedom in a test, the more the distribution curves toward a normal distribution.
Critical Value
The critical value in a chi-square test serves as the threshold that determines whether to accept or reject the null hypothesis. It is obtained from a chi-square distribution table, depending on the degrees of freedom and the desired level of significance.
In this specific exercise, with 2 degrees of freedom and a 伪 = 0.05 level of significance, the critical value is 5.99. This means that to reject the null hypothesis, the calculated 蠂^2 value must exceed this critical value. Since our observed 蠂^2 is 4.63, it is less than the critical value, leading us to not reject the null hypothesis.
Level of Significance
The level of significance ( 伪 ) is a threshold that defines the probability of rejecting the null hypothesis when it is actually true. Essentially, it quantifies the risk of making a Type I error 鈥� falsely concluding that the null hypothesis is incorrect. In this exercise, a 0.05 level of significance means there is a 5% risk that any decision to reject the null hypothesis might be wrong.
Choosing a level of significance depends on the context of the study and how much risk of error is acceptable. In scientific studies, a 伪 = 0.05 is commonly used because it strikes a balance between being too lenient and too strict. If the calculated 蠂^2 falls below the critical value, as it does here (4.63 < 5.99), we do not reject the null hypothesis, staying within this acceptable risk range.

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Most popular questions from this chapter

Drivers born under the astrological sign of Capricorn are the worst drivers in Australia, according to an article that appeared in the Australian newspaper The Mercury (October 26,1998 ). This statement was based on a study of insurance claims that resulted in the following data for male policyholders of a large insurance company. $$ \begin{array}{lc} \text { Astrological Sign } & \begin{array}{c} \text { Number of } \\ \text { Policyholders } \end{array} \\ \hline \text { Aquarius } & 35,666 \\ \text { Aries } & 37,926 \\ \text { Cancer } & 38,126 \\ \text { Capricorn } & 54,906 \\ \text { Gemini } & 37,179 \\ \text { Leo } & 37,354 \\ \text { Libra } & 37,910 \\ \text { Pisces } & 36,677 \\ \text { Sagittarius } & 34,175 \\ \text { Scorpio } & 35,352 \\ \text { Taurus } & 37,179 \\ \text { Virgo } & 37,718 \\ \hline \end{array} $$ a. Assuming that it is reasonable to treat the male policyholders of this particular insurance company as a random sample of male insured drivers in Australia, are the observed data consistent with the hypothesis that the proportion of male insured drivers is the same for each of the 12 astrological signs? b. Why do you think that the proportion of Capricorn policyholders is so much higher than would be expected if the proportions are the same for all astrological signs? c. Suppose that a random sample of 1000 accident claims submitted to this insurance company is selected and each claim classified according to the astrological sign of the driver. (The accompanying table is consistent with accident rates given in the article.) $$ \begin{array}{lc} & \text { Observed Number } \\ \text { Astrological Sign } & \text { in Sample } \\ \hline \text { Aquarius } & 85 \\ \text { Aries } & 83 \\ \text { Cancer } & 82 \\ \text { Capricorn } & 88 \\ \text { Gemini } & 83 \\ \text { Leo } & 83 \\ \text { Libra } & 83 \\ \text { Pisces } & 82 \\ \text { Sagittarius } & 81 \end{array} $$ $$ \begin{array}{lc} & \text { Observed Number } \\ \text { Astrological Sign } & \text { in Sample } \\ \hline \text { Scorpio } & 85 \\ \text { Taurus } & 84 \\ \text { Virgo } & 81 \\ \hline \end{array} $$ Test the null hypothesis that the proportion of accident claims submitted by drivers of each astrological sign is consistent with the proportion of policyholders of each sign. Use the given information on the distribution of policyholders to compute expected frequencies and then carry out an appropriate test.

The paper "Credit Card Misuse, Money Attitudes, and Compulsive Buying Behavior: Comparison of Internal and External Locus of Control Consumers" (College Student Journal [2009]: \(268-275\) ) describes a study that surveyed a sample of college students at two midwestern public universities. Based on the survey responses, students were classified into two "locus of control" groups (internal and external) based on the extent to which they believe that they control what happens to them. Those in the internal locus of control group believe that they are usually in control of what happens to them, whereas those in the external locus of control group believe that it is usually factors outside their control that determines what happens to them. Each student was also classified according to a measure of compulsive buying. The resulting data are summarized in the accompanying table. Can the researchers conclude that there are an association between locus of control and compulsive buying behavior? Carry out a \(X^{2}\) test using \(\alpha=.01\). Assume it is reasonable to regard the sample as representative of college students at midwestern public universities. $$ \begin{array}{l|ccc} & & \text { Locus of Control } \\ \hline & \text { Internal } & \text { External } \\ \hline \text { Compulsive } & \text { Yes } & 3 & 14 \\ \text { Buyer? } & \text { No } & 52 & 57 \\ \hline \end{array} $$

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

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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