Question

question 1
1 pts

after surgery a patients blood volume is often depleted. in one study, the total circulating volume of blood plasma was measured for each patient immediately after surgery. after infusion of a \plasma expander\ into the bloodstream, the plasma volume was measured again and the increase in plasma volume (ml) was calculated. two of the plasma expanders used were albumin (25 patients) and polygelatin (14 patients). the following table reports the increase in plasma volume (samuels, witmer, and schaffner (1989)).

	albumin	polygelatin
\\(\bar{x}\\)	490	240
s	60	30

which option correctly states the null and alternative hypotheses for testing whether there is a significant difference in the average plasma volume between the two types of plasma expanders?

\\(h_0: \mu_{\text{albumin}} - \mu_{\text{polygelatin}} = 0\\) vs. \\(h_1: \mu_{\text{albumin}} - \mu_{\text{polygelatin}} \
eq 0\\)
\\(h_0: \mu_{\text{albumin}} - \mu_{\text{polygelatin}} \ge 0\\) vs. \\(h_1: \mu_{\text{albumin}} - \mu_{\text{polygelatin}} < 0\\)
\\(h_0: \mu_{\text{albumin}} - \mu_{\text{polygelatin}} \le 0\\) vs. \\(h_1: \mu_{\text{albumin}} - \mu_{\text{polygelatin}} > 0\\)

Explanation:

⚡ Using what you learned: Hypothesis Testing

Step 1: Identify the research question

The goal is to test "whether there is a significant difference in the average plasma volume between the two types of plasma expanders."

A test for "a significant difference" does not specify a direction (i.e., it does not ask if one is specifically larger or smaller than the other). This requires a two-tailed test.

Step 2: Formulate the hypotheses

For a two-tailed test comparing two population means, \( \mu_{\text{Albumin}} \) and \( \mu_{\text{Polygelatin}} \):

• Null Hypothesis (\( H_0 \)): There is no difference between the two means.

$$ H_0 : \mu_{\text{Albumin}} - \mu_{\text{Polygelatin}} = 0 $$

• Alternative Hypothesis (\( H_1 \)): There is a difference between the two means.

$$ H_1 : \mu_{\text{Albumin}} - \mu_{\text{Polygelatin}} eq 0 $$

Answer:

\( H_0 : \mu_{\text{Albumin}} - \mu_{\text{Polygelatin}} = 0 \text{ vs. } H_1 : \mu_{\text{Albumin}} - \mu_{\text{Polygelatin}}
eq 0 \)