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Chapter 14: Problem 2

Use Table 5 in Appendix I to find the value of \(\chi^{2}\) with the following area \(\alpha\) to its right: a. \(\alpha=.05, d f=3\) b. \(\alpha=.01, d f=8\)

Short Answer

Expert verified
Question: What are the values of \(\chi^{2}\) for the following: a. \(\alpha=.05, df=3\) b. \(\alpha=.01, df=8\) Answer: a. \(\chi^{2} = 7.81\) b. \(\chi^{2} = 20.09\)

Step by step solution

01

Find the row corresponding to the degrees of freedom (\(df\)) given in the problem. In this case, our given \(df = 3\). So, locate the row corresponding to 3 degrees of freedom in Table 5 in Appendix I. #Step 2: Locate the area to the right in the table#

Locate the column corresponding to \(.05\) in Table 5. This column will have the chi-square values (\(\chi^{2}\)) associated with an area of \(.05\) to their right. #Step 3: Find the value of \(\chi^{2}\) at the intersection of row and column#
02

The intersection of the row corresponding to 3 degrees of freedom and the column corresponding to \(.05\) will give you the value of \(\chi^{2}\). In this case, the value of \(\chi^{2}\) is 7.81. #b. \(\alpha=.01, d f=8\)# #Step 1: Locate the degrees of freedom in the table#

Find the row corresponding to the given degrees of freedom (\(df\)). In this case, our given \(df = 8\). So, locate the row corresponding to 8 degrees of freedom in Table 5 in Appendix I. #Step 2: Locate the area to the right in the table#
03

Locate the column corresponding to \(.01\) in Table 5. This column will have the chi-square values (\(\chi^{2}\)) associated with an area of \(.01\) to their right. #Step 3: Find the value of \(\chi^{2}\) at the intersection of row and column#

The intersection of the row corresponding to 8 degrees of freedom and the column corresponding to \(.01\) will give you the value of \(\chi^{2}\). In this case, the value of \(\chi^{2}\) is 20.09. So, the values of \(\chi^{2}\) for the given problems are: a. \(\chi^{2} = 7.81\) for \(\alpha=.05, df=3\). b. \(\chi^{2} = 20.09\) for \(\alpha=.01, df=8\).

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Most popular questions from this chapter

A peony plant with red petals was crossed with another plant having streaky petals. A geneticist states that \(75 \%\) of the offspring from this cross will have red flowers. To test this claim, 100 seeds from this cross were collected and germinated, and 58 plants had red petals. Use the chi-square goodness-of- fit test to determine whether the sample data confirm the geneticist's prediction.

Suppose you are interested in following two independent traits in snap peas- seed texture \((\mathrm{S}=\) smooth, \(\mathrm{s}=\) wrinkled \()\) and seed color \((\mathrm{Y}=\) yellow, \(\mathrm{y}=\) green \()-\) in a second-generation cross of heterozygous parents. Mendelian theory states that the number of peas classified as smooth and yellow, wrinkled and yellow, smooth and green, and wrinkled and green should be in the ratio \(9: 3: 3: 1 .\) Suppose that 100 randomly selected snap peas have \(56,\) \(19,17,\) and 8 in these respective categories. Do these data indicate that the 9: 3: 3: 1 model is correct? Test using \(\alpha=.01\)

You can use a goodness-of-fit test to determine whether all of the criteria for a binomial experiment have actually been met in a given application. Suppose that an experiment consisting of four trials was repeated 100 times. The number of repetitions on which a given number of successes was obtained is recorded in the table: Estimate \(p\) (assuming that the experiment was binomial), obtain estimates of the expected cell frequencies, and test for goodness of fit. To determine the appropriate number of degrees of freedom for \(X^{2}\), note that \(p\) was estimated by a linear combination of the observed frequencies.

Suppose that a consumer survey summarizes the responses of \(n=307\) people in a contingency table that contains three rows and five columns. How many degrees of freedom are associated with the chi-square test statistic?

Parents who are concerned about public school environments and curricula are turning to homeschooling in order to control the content and atmosphere of the learning environments of their children. Although employment as a public school teacher requires a bachelor's degree in education or a subject area, the educational background of homeschool teachers is quite varied. The educational background of a sample of \(n=500\) parents involved in homeschooling their children in 2003 are provided in the first table that follows, along with the corresponding percentages for parents who homeschooled in \(1999 .\) The education levels for U.S. citizens in general are given in the second 16 table. a. Is there a significant change in the educational backgrounds of parents who homeschooled their children in 2003 compared with \(1999 ?\) Use \(\alpha=.01\) b. If there is a significant change in the educational backgrounds of these parents, how would you describe that change? c. Using the second table, can we determine if homeschool teachers have the same educational backgrounds as the U.S. population in general? If not, which groups are underrepresented and which are overrepresented?
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