Chapter 7. Hypothesis Testing with One Sample

Chapter 7: Hypothesis Testing with One Sample 227b. For α = 0.05, the critical values found on Table A-2 are z = -1.96 and z = 1.96. Also, k = degrees offreedom = n – 1 = 149. Calculating the left critical value for a χ 2 , we find( ) 2 = 116.643.χ 2 = 1 2 (z + 2k −1) 2 = 1 −1.96 + 2 ×149 −12Calculating the right critical value for χ 2 , we findχ 2 = 1 2 (z + 2k −1) 2 = 1 1.96 + 2 ×149 −12( ) 2 = 184.19914. Finding Critical Values for χ 2a. For α = 0.05, the critical values found on Table A-2 are z = -1.96 and z = 1.96. Also, k = degrees offreedom = n – 1 = 100. Calculating the left critical value for a χ 2 , we findχ 2 = k 1− 29k + z 29k3= 100 1−29 ×100 −1.96 29 ×100Calculating the right critical value for χ 2 , we findχ 2 = k 1− 29k + z 29k3= 100 1−29 ×100 +1.96 29 ×10033= 74.216.= 129.565The values from Table A-4 are 74.222 and 129.561. The values found using the formula are extremelyaccurate (each is off by less than 0.01).b. For α = 0.05, the critical values found on Table A-2 are z = -1.96 and z = 1.96. Also, k = degrees offreedom = n – 1 = 149. Calculating the left critical value for a χ 2 , we findχ 2 = k 1− 29k + z 29k3= 149 1−29 ×149 −1.96 29 ×149Calculating the right critical value for χ 2 , we findχ 2 = k 1− 29k + z 29k3= 149 1−29 ×149 +1.96 29 ×14933= 117.093.= 184.690Review Exercises1. a. We can conclude that most adult Americans are opposed to estate taxes, but we need to be careful. Withsuch a large sample size, a sample proportion that is only slightly higher than 0.50 would result in thatconclusion.b. This drug should not be used. The actual mean weight loss is insignificant, though it does have statisticalsignificance.c. The 0.001 significance level would be most convincing. However, I would prefer the 0.01 significancelevel, as it is a reasonable threshold without being unrealistic.d. There is not sufficient evidence to warrant the rejection of the claim that the volume of cold medicine isdifferent than 12 oz.e. “rejecting a true null hypothesis.”2. Identifying Hypotheses and Distributionsa. H 1 : µ < 10,000. Since the sample size is larger than 30 (n = 750) and the population standard deviation isunknown, the sampling distribution is the Student t distribution.b. H 1 : σ > 1.8. We do not know the distribution of the population, and so we cannot use the χ 2 distribution.However, had we known the distribution was normally distributed, we would use the distribution, sa weare performing a hypothesis test regarding the population standard deviation.c. H 1 : p > 0.50. Since np = 500 × 0.50 = 250 and nq = 500 × 0.50 = 250, the sampling distribution is thenormal distribution.d. H 1 : µ ≠ 100. Since the sample size is larger than 30 (n = 150) and the population standard deviation isknown, the sampling distribution is the normal distribution.