Question

Question 18, c d e

18. Medical: Hemoglobin Count Let x be a random variable that represents hemoglobin count (HC) in grams per 100 milliliters of whole blood. Then x has a distribution that is approximately normal, with a population mean of about 14 for healthy adult women (see reference in Problem 17). Suppose female patient has taken 10 laboratory od tests during the past year. The HC data sent to the patient's doctor are 2 that a 15 18 16 19 14 12 14 17 15 i11 i. Use a calculator with sample verify that x = 15.1 and ii. Does this information indicate that the population average HC for this patient is higher than 14? Use a = 0.01. mean and sample standard deviation keys to 2.51. S (c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not given in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conser- vative" answer.

Answer 1

The sample size is n = 10. The provided sample data along with the data required to compute the sample mean X and sample variance s2 are shown in the table below: x2 X 15 225 18 324 16 256 19 361 14 196 12 144 14 196 17 289 15 225 11 121 Sum 151 2337 The sample mean X is computed as follows:

7 151 1 X = Xi 15.1 10 n =W

The sample variance s2 is 2 n 7 1512 1 1 1 2337 = 6.322 10 1 10 1 n i=1 i=1 Therefore, the sample standard deviation s is 8 Vs2 =6.322 = 2.514

The provided sample mean is X = 15.1 and the sample standard deviation is s = 2.514, and the sample size is n = 10. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho. μ< 14 Ha: u 14 > This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used. (2) Rejection Region Based on the information provided, the significance level is a 0.01, and the critical value for a right-tailed test is te = 2.821. The rejection region for this right-tailed test is R = t :t > 2.821 (3) Test Statistics The t-statistic is computed as follows: X - Ho 15.1 14 1.384 8/ n 2.514/10

(4) Decision about the null hypothesis 1.384 < te = 2.821, it is then concluded that the null hypothesis is not Since it is observed that t rejected Using the P-value approach: The p-value is p 0.0999, and since p = 0.0999 > 0.01, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean u is greater than 14, at the 0.01 significance level. Confidence Interval The 99% confidence interval is 12.516 < u < 17.684