Question

Suppose a researcher wants to perform an hypothesis test to decide whether the mean systolic blood pressure for adults aged 20-29 is lower than the mean systolic blood pressure for adults aged 30-39. State the null and alternative hypothesis for the hypothesis test. Discuss the basic strategy for performing this hypothesis test, based on independent samples.

Answer 1

Let Group 1 be the adults aged 20-29 and Group 2 be the adults aged 30-39 and $\small \mu_1$ and $\small \mu_2$ be the mean systolic blood pressures of the two groups respectively. Then, we want to test if $\small \mu_1 < \mu_2$ . Thus, the hypotheses are given by:

$\small \\ H_0 : \mu_1 = \mu_2 \\ H_1 : \mu_1 < \mu_2$

Now, to test the hypothesis using independent samples we could collect samples from the two groups and calculate the sample mean and sample standard deviation for both the groups. Then we know that, under H₀:

$\small \frac{(\bar{X}_1-\bar{X}_2 )- (\mu_1-\mu_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S^2_2}{n_2}}} \sim t_n$

where

$\small n = \frac{(A+B)^2}{\frac{A^2}{n_1-1} + \frac{B^2}{n_2-1}}$

where

$\small A = \frac{S_1^2}{n_1} \ \ \ \& \ \ \ \ B = \frac{S_2^2}{n_2}$

We can use the above test statistic to test the given hypothesis. When the value of the test statistic is too negative, we reject the null hypothesis.

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