Question

the owner of a fitness watch would like to determine if the mean number of steps he takes per day differs from the recommended 10,000 steps per day, using $\alpha = 0.01$. he selects a random sample of 50 days with the intention of testing the hypotheses $h_{0}:\mu = 10,000$ steps versus $h_{a}:\mu \
eq 10,000$ steps where $\mu =$ the true mean number of steps taken per day.
which of the following values of the alternative hypothesis would yield the greatest power to reject the null hypothesis?
$\bigcirc \mu = 9,000$
$\bigcirc \mu = 9,500$
$\bigcirc \mu = 10,000$
$\bigcirc \mu = 10,500$

Explanation:

Step1: Understand the concept of power in hypothesis testing

Power is the probability of correctly rejecting the null hypothesis \(H_0\) when the alternative hypothesis \(H_a\) is true. The further the true value of the parameter (in this case \(\mu\)) is from the hypothesized value under \(H_0\) (\(\mu_0 = 10000\)), the greater the power.

Step2: Calculate the distance from \(\mu_0 = 10000\) for each option

• For \(\mu=9000\): \(|10000 - 9000|=1000\)
• For \(\mu = 9500\): \(|10000 - 9500|=500\)
• For \(\mu=10000\): \(|10000 - 10000|=0\) (this is the null hypothesis value, so no power to reject \(H_0\) as \(H_0\) is True)
• For \(\mu=10500\): \(|10000 - 10500|=500\)

Since \(1000>500 = 500>0\)

Answer:

\(\mu = 9000\)