5. To investigate whether people suffering from hypertension have different average daily salt intake than the...

 

Transcribed Image Text:

5. To investigate whether people suffering from hypertension have different average daily salt intake than the general population, the average daily salt intake was measured for a sample of 10 hypertensives. The data are as follows: 92.8, 54.8, 51.6, 61.7, 50.8, 84.5, 34.7, 62.2, 11.0, 39.1 These data are not normally distributed. However, the natural logarithm of salt intake is approximately normally distributed. The sample mean of the log* salt intake is a 3.871 with a sample sd of s = .599. It is known that the average salt intake in the general population (on log scale) is 2.2 with sd 1.1. The question of interest here is, Do hypertensives have elevated salt intake relative to the general population? a. Suppose we are willing to assume that the population sd o for hypertensives is the same as in the general population. That is, suppose o = 1.1. Let H1 denote the population mean natural log salt intake for hypertensives. * All logs are natural logs in this course unless stated otherwise. That is, I write log(x) for the natural log of x. 0.01, test the hypothesis Ho : µ1 = 2.2 assuming a two-sided alternative. Report the test statistic, its null distribution, p-value and the conclusion. The null hypothesis here is in terms of the population means for log salt intake among hypertensives and in the general population, but does your conclusion apply to the population means for salt intake as well? At level a = b. Now suppose that we are not willing to assume that the population sd among hypertensives is the same as that among the general population. Therefore, we must estimate o rather than assume we know it. Repeat the test from part (a) under this scenario. c. Suppose I am willing to assume that the population mean log salt intake among hypertensives is not less than in the general population and I wish to investigate whether it is more than that of the general population. As in part (b), do not assume that the population SD is known. State the appropriate null and alternative hypotheses and repeat the test of part (b). How does your p-value compare to that of part (b). d. Now suppose that little is known about salt intake in the general population. So the average daily salt intake was measured for each member of a random sample of 15 people without hypertension. The data obtained from this sample are as follows: 1.6, 20.4, 44.9, 4.2, 2.0, 1.1, 31.8, 33.3, 14.8, 2.8, 10.9, 114.7, 3.0,3.8, 6.1 At level a = 0.01, test the hypothesis that µ2, the population mean log salt intake among non-hypertensives, is the same as that of hypertensives versus an alternative that they are unequal. Do not assume that the pop- ulation SDs are equal or known. As always, state the test statistic, its null distribution, p-value, and your conclusion. e. In part (d), you tested Ho : H1 = H2 or, equivalently, Ho : H1 - H2 using a t test. Now obtain and interpret the corresponding 99% confidence interval for µi - H2. 5. To investigate whether people suffering from hypertension have different average daily salt intake than the general population, the average daily salt intake was measured for a sample of 10 hypertensives. The data are as follows: 92.8, 54.8, 51.6, 61.7, 50.8, 84.5, 34.7, 62.2, 11.0, 39.1 These data are not normally distributed. However, the natural logarithm of salt intake is approximately normally distributed. The sample mean of the log* salt intake is a 3.871 with a sample sd of s = .599. It is known that the average salt intake in the general population (on log scale) is 2.2 with sd 1.1. The question of interest here is, Do hypertensives have elevated salt intake relative to the general population? a. Suppose we are willing to assume that the population sd o for hypertensives is the same as in the general population. That is, suppose o = 1.1. Let H1 denote the population mean natural log salt intake for hypertensives. * All logs are natural logs in this course unless stated otherwise. That is, I write log(x) for the natural log of x. 0.01, test the hypothesis Ho : µ1 = 2.2 assuming a two-sided alternative. Report the test statistic, its null distribution, p-value and the conclusion. The null hypothesis here is in terms of the population means for log salt intake among hypertensives and in the general population, but does your conclusion apply to the population means for salt intake as well? At level a = b. Now suppose that we are not willing to assume that the population sd among hypertensives is the same as that among the general population. Therefore, we must estimate o rather than assume we know it. Repeat the test from part (a) under this scenario. c. Suppose I am willing to assume that the population mean log salt intake among hypertensives is not less than in the general population and I wish to investigate whether it is more than that of the general population. As in part (b), do not assume that the population SD is known. State the appropriate null and alternative hypotheses and repeat the test of part (b). How does your p-value compare to that of part (b). d. Now suppose that little is known about salt intake in the general population. So the average daily salt intake was measured for each member of a random sample of 15 people without hypertension. The data obtained from this sample are as follows: 1.6, 20.4, 44.9, 4.2, 2.0, 1.1, 31.8, 33.3, 14.8, 2.8, 10.9, 114.7, 3.0,3.8, 6.1 At level a = 0.01, test the hypothesis that µ2, the population mean log salt intake among non-hypertensives, is the same as that of hypertensives versus an alternative that they are unequal. Do not assume that the pop- ulation SDs are equal or known. As always, state the test statistic, its null distribution, p-value, and your conclusion. e. In part (d), you tested Ho : H1 = H2 or, equivalently, Ho : H1 - H2 using a t test. Now obtain and interpret the corresponding 99% confidence interval for µi - H2.

Answer rating: 100% (QA)

d In this case first we have to transform the given data ... View the full answer