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Chapter 15: Problem 8

Packages of mixed nuts made by a certain company contain four types of nuts. The percentages of nuts of Types \(1,2,3,\) and 4 are advertised to be \(40 \%, 30 \%, 20 \%,\) and \(10 \%,\) respectively. A random sample of nuts is selected, and each one is categorized by type. a. If the sample size is 200 and the resulting test statistic value is \(X^{2}=19.0,\) what conclusion would be appropriate for a significance level of \(0.001 ?\) b. If the random sample had consisted of only 40 nuts, would you use the chi- square goodness-of-fit test? Explain your reasoning.

Short Answer

Expert verified
For a sample of 200 nuts, 1,000 significance level, and a test statistic value of 19.0, the conclusion is that the proportions of nuts are not as advertised. Chi-square goodness-of-fit test is not appropriate for a sample size of 40 because the expected frequency of at least one type of nut is less than 5.

Step by step solution

01

Calculate expected frequencies

The expected frequency for each type of nut can be calculated by multiplying the sample size with the percentage of each type of nut. Therefore for Type 1 nuts, the expected frequency is \(40\% * 200 = 80\). Following the same calculation for Type 2, Type 3 and Type 4 gives 60, 40 and 20 respectively.
02

Conduct Chi-square goodness-of-fit test

Now, the computed value of the test statistic, \(X^{2}=19.0,\) compares the observed frequencies in the sample with the expected frequencies mentioned above. The appropriate conclusion is found by comparing the calculated test statistic with the critical value from the chi-square distribution with degrees of freedom \(df = n - 1\), where \(n\) is the number of categories (in this case, the types of nuts). Since we have 4 types of nuts, \(df = 4 - 1 = 3\). If the calculated test statistic is greater than the critical value, we reject the null hypothesis, indicating that the proportions of nuts are not as advertised. Using a standard chi square table, a \(X^2\) value of 19 exceeds the critical value in the \(X^2\) distribution for \(df = 3\) at the .001 significance level, indicating that the null hypothesis should be rejected.
03

Assess Appropriateness of the Test for Smaller Sample size

For a sample size of 40, using the chi-square goodness-of-fit test may not be appropriate since the test requires that expected frequencies should be at least 5 for all categories. In this case, if we calculate the expected frequencies for a sample size of 40, the expected frequency for the 4th type of nut would be \(10\% * 40 = 4\), which is less than 5. So, we cannot use the chi-square goodness-of-fit in this case.

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Most popular questions from this chapter

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 \(p_{1}=0.25, p_{2}=0.50,\) and \(p_{3}=0.25 .\) 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 \(\alpha=0.05\). 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)?

A particular cell phone case is available in a choice of four different colors. A store sells all four colors. To test the hypothesis that sales are equally divided among the four colors, 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 \(0.05 ?\) b. What conclusion would be appropriate at significance level 0.01 if \(X^{2}=15.3 ?\) c. If there were six different colors rather than just four, what would you conclude if \(X^{2}=13.7\) and a test with \(\alpha=0.05\) was used?

Give an example of a situation where it would be appropriate to use a chi- square test of independence. Describe the population that would be sampled and the two variables that would be recorded.

What is the approximate \(P\) -value for the following values of \(X^{2}\) and df? a. \(X^{2}=14.44, \mathrm{df}=6\) b. \(X^{2}=16.91, \mathrm{df}=9\) c. \(X^{2}=32.32, \mathrm{df}=20\)

Does viewing angle affect a person's ability to tell the difference between a female nose and a male nose? This important (?) research question was examined in the article "You Can Tell by the Nose: Judging Sex from an Isolated Facial Feature" (Perception [1995]: \(969-973\) ). Eight Caucasian males and eight Caucasian females posed for nose photos. The article states that none of the volunteers wore nose studs or had prominent nasal hair. Each person placed a black Lycra tube over his or her head in such a way that only the nose protruded through a hole in the material. Photos were then taken from three different angles: front view, three-quarter view, and profile. These photos were shown to a sample of undergraduate students. Each student in the sample was shown one of the nose photos and asked whether it was a photo of a male or a female. The response was classified as either correct or incorrect. The accompanying table was constructed using summary values reported in the article. Is there evidence that the proportion of correct sex identifications differs for the three different nose views?
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