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Explain the difference between the observed and expected frequencies for a goodness-of-fit test.

To make a goodness-of-fit test, what should be the minimum expected frequency for each category? What are the alternatives if this condition is not satisfied?

Explain how the expected frequencies for cells of a contingency table are calculated in a test of independence or homogeneity.

A sample of 21 observations selected from a normally distributed population produced a sample variance of \(1.97 .\) a. Write the null and alternative hypotheses to test whether the population variance is greater than \(1.75\). b. Using \(\alpha=.025\), find the critical value of \(\chi^{2}\). Show the rejection and nonrejection regions on a chi-square distribution curve. c. Find the value of the test statistic \(\chi^{2}\). d. Using a \(2.5 \%\) significance level, will you reject the null hypothesis stated in part a?

According to a Gallup poll whose results were reported on October 22, 2013, American's views on legalizing marijuana are changing. In that survey, American adults were asked whether marijuana should be legalized in America. Suppose in a recent survey, 600 Americans were randomly selected from each of the four age groups listed in the table below. The frequencies of the responses for various age groups are listed in this table assuming that every person included in the survey responded yes or no. $$ \begin{array}{lll} \hline & \text { Yes } & \text { No } \\ \hline 18 \text { to } 29 & 402 & 198 \\ 30 \text { to } 49 & 372 & 228 \\ 50 \text { to } 64 & 336 & 264 \\ 65+ & 270 & 330 \\ \hline \end{array} $$ Test at a \(1 \%\) significance level whether the proportion of Americans who support legalizing marijuana is the same for each of the age groups.

A random sample of 100 persons was selected from each of four regions in the United States. These people were asked whether or not they support a certain farm subsidy program. The results of the survey are summarized in the following table. $$ \begin{array}{lccc} \hline & \text { Favor } & \text { Oppose } & \text { Uncertain } \\ \hline \text { Northeast } & 56 & 33 & 11 \\ \text { Midwest } & 73 & 23 & 4 \\ \text { South } & 67 & 28 & 5 \\ \text { West } & 59 & 35 & 6 \\ \hline \end{array} $$ Using a \(1 \%\) significance level, test the null hypothesis that the percentages of people with different opinions are similar for all four regions.

Each of five boxes contains a large (but unknown) number of red and green marbles. You have been asked to find if the proportions of red and green marbles are the same for each of the five boxes. You sample 50 times, with replacement, from each of the five boxes and observe \(20,14,23,30\), and 18 red marbles, respectively. Can you conclude that the five boxes have the same proportion of red and green marbles? Use a . 05 level of significance.