Question

1. The distribution of diastolic blood pressures for the population of female diabetics between the ages of 30 and 34 has an unknown mean and standard deviation.  A sample of 10 diabetic women is selected; their mean diastolic blood pressure is 84 mm Hg. We want to determine whether the diastolic blood pressure of female diabetics are different from the general population of females in this age group, where the mean μ = 74.4 mmHg and standard deviation σ = 9.1 mm Hg.  Diastolic blood pressure is normally distributed.    
a) Create a two-sided 95% confidence interval to determine whether diabetic women have a different mean diastolic blood pressure compared to the general population.   
b) Now, conduct a two-sided hypothesis test at the α = 0.05 level of significance to determine whether diabetic women have a different mean diastolic blood pressure compared to the general population.  Use both critical value and p-value methods.
For either method, would your conclusion have been different if you had chosen α = 0.01 instead of α = 0.05?

Answer 1

a) At 95% confidence interval the critical value is z_0.025 = 1.96

The 95% confidence interval is

+/- z_0.025 *

= 84 +/- 1.96 * 9.1/

= 84 +/- 5.64

= 78.36, 89.64

b) H₀: = 74.4

   H₁: 74.4

The test statistic z = ()/()

                           = (84 - 74.4)/( 9.1/)

                           = 3.34

At = 0.05, the critical value is +/- z_0.025 = 1.96

Since the test statistic value is greater than the positive critical value (3.34 > 1.96), so we should reject the null hypothesis.

P-value = 2 * P(Z > 3.34)

            = 2 * (1 - P(Z < 3.34))

            = 2 * (1 - 0.9996)

            = 0.0008

Since the P-value is less than the significance level(0.0008 < 0.05), so we should reject the null hypothesis.

At 0.05 significance level, there is sufficient to conclude that the diastolic blood pressure of female diabetics are different from the general population of females in this age group.

At = 0.01, since the P-value is less than , so we should reject the null hypothesis.

No, the conclusion would not have been different if we had chosen = 0.01 instead of = 0.05.