1. A number cube is rolled 35 times and the results are recorded in the table. Number Rolled 1 2 3 4 5 6 7 Frequency 1 5 7 9 7 5 1 A goodness-of-fit chi-square test is to be used to test the null hypothesis that the cube is fair. At a significance level of ? = 0.05, what is the value of chi-square and the appropriate conclusion? ??2 = 13.6; fail to reject the null hypothesis ??2 = 11.2; fail to reject the null hypothesis ??2 = 27.2; fail to reject the null hypothesis ??2 = 27.2; reject the null hypothesis ??2 = 20; reject the null hypothesis 2. Which of the following correctly describes how the width of a confidence interval for a population mean changes when the population standard deviation is known? There is no change if just the sample size increases. The interval widens if the sample size decreases and confidence level stays the same. The interval narrows if the sample size decreases and confidence level stays the same. The interval widens if the sample size increases and the confidence level stays the same. The interval widens if the sample size stays the same and confidence level decreases.

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Solution:The null and alternative hypotheses are:\mathrm{HO} : The cube is fairHa: The cube is not fair.The test statistic is:\chi^{2}=\sum \frac{(O-E)^{2}}{E}Therefore, the test statistic is:\chi^{2}=\sum \frac{(O-E)^{2}}{E}=11.2The p-value is:p-\text { value }=P\left(\chi^{2}>11.2\right)=0.0824The excel function to find the \mathrm{p}-value is:= CHIDIST (11.2,6)Where:11.2 is the test statistic and 6 is the dfSince the p-value is greater than the significance level 0.05 , we, therefore, fail to reject the null hypothesis.A goodness-of-fit chi-square test is to be used to test the null hypothesis that the cube is fair. At a significance level of \alpha=0.05, what is the value of chi-square and the appropriate conclusion?\chi^{2}=13.6; fail to reject the null hypothesisx^{2}=11.2; fail to reject the null hypothesis\chi^{2}=27.2; fail to reject the null hypothesis\chi^{2}=27.2 ; reject the null hypothesisx^{2}=20; reject the null hypothesis ...