Question

a knee replacement currently has an average short-term recovery time of 90 days. a physical therapist has developed a new therapy program that is designed to shorten the recovery time and would like to conduct a hypothesis test to verify its effectiveness.

what are the appropriate null and alternative hypotheses?

\\( h_0 : \mu = 90 \\)
\\( h_1 : \mu \
eq 90 \\)

\\( h_0 : \mu = 90 \\)
\\( h_1 : \mu > 90 \\)

\\( h_0 : \mu = 90 \\)
\\( h_1 : \mu < 90 \\)

Explanation:

Step1: Understand Hypothesis Testing

The null hypothesis ($H_0$) is a statement of no effect or no difference. Here, the current average recovery time is 90 days, so $H_0: \mu = 90$.

Step2: Determine Alternative Hypothesis

The new therapy is designed to shorten recovery time, meaning we expect the average recovery time ($\mu$) to be less than 90. So the alternative hypothesis ($H_1$) is $H_1: \mu < 90$.

Step3: Evaluate Options

• First option: $H_1: \mu

eq 90$ tests for a difference (two - tailed), not a decrease.

• Second option: $H_1: \mu > 90$ tests for an increase, which is opposite of what we want.
• Third option: Matches our reasoning with $H_0: \mu = 90$ and $H_1: \mu < 90$.

Answer:

$H_0: \mu = 90$
$H_1: \mu < 90$ (the third option in the given choices)