Question

question 1

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)).

\

$$\begin{tabular}{|c|c|c|} \\hline & albumin & polygelatin \\\\ \\hline n & 25 & 14 \\\\ \\hline \\bar{x} & 490 & 240 \\\\ \\hline s & 60 & 30 \\\\ \\hline \\end{tabular}$$

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 objective is to test whether there is a significant difference in the average plasma volume increase between the two types of plasma expanders (Albumin and Polygelatin).

A "significant difference" does not specify a direction (i.e., it does not ask if one is strictly greater than or less than the other). This indicates a two-tailed test.

Step 2: Formulate the hypotheses

Let \( \mu_{\text{Albumin}} \) be the population mean increase in plasma volume for Albumin, and \( \mu_{\text{Polygelatin}} \) be the population mean increase in plasma volume for Polygelatin.

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

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

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

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

Answer:

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