Chapter 7. Hypothesis Testing with One Sample

Chapter 7: Hypothesis Testing with One Sample 225There is sufficient evidence to warrant the rejection of the claim that the population standard deviation of birthweights of male babies born to mothers taking a special vitamin supplement is 0.470 kg.9. Is the New Machine Better?The claim is that the population standard deviation of cold medicine poured by the new machine is less thanthat of the old machine, which was 0.15 oz., so this is a left-tailed test. The sample size is n = 71 making thedegrees of freedom df = 70. The significance level is 0.05. The sample standard deviation is s = 0.12 oz. Wemove on to the test.H 0 : σ = 0.15H 1 : σ < 0.1570 ⋅ 0.1220.15 2 = 44.82(n −1)sThe test statistic is χ 2 = =σ 2In a left-tailed test at the 0.05 significance level with df = 70, the critical value is χ 2 = 51.739.In the row for 70 degrees of freedom, the test statistic lies between 43.275 and 45.442, so the P-value isbetween 0.005 and 0.01.We reject the null hypothesis.The sample data support the claim that the population standard deviation of cold medicine poured by the newmachine is less than that of the old machine. Its purchase should be considered.10. Systolic Blood Pressure for WomenWe use the pre-exercise, no stress systolic blood pressure for this hypothesis test. The claim is that thepopulation standard deviation of systolic blood pressure for women is 23.4, so this is a two-tailed test. Thesample size is n = 40 making the degrees of freedom df = 39. The significance level is not mentioned, so weuse 0.05. Calculating the sample standard deviation from the data givess = nΣ(x 2 ) − (Σx) 240 ⋅ 502,488 − (4432)2= = 17.114n(n −1)40 ⋅ 39We move on to the test.H 0 : σ = 23.4H 1 : σ 23.4 ≠22(n −1)s 39 ⋅17.114The test statistic is χ 2 = = = 20.861σ 2 23.4 2The degrees of freedom df = 39 is not displayed in Table A-4, so we use df = 40. In a two-tailed test at the 0.05significance level with df = 40, the critical values are χ 2 = 24.433 and χ 2 = 59.342.In the row for 40 degrees of freedom, the test statistic lies between 20.707 and 22.164, so the P-value isbetween 0.01 and 0.02.We reject the null hypothesis.There is sufficient evidence to warrant rejection of the claim that the population standard deviation of systolicblood pressure for women is 23.4.