Questions: The director of research and development is testing a new medicine. She wants to know if there is evidence at the 0.05 level that the medicine relieves pain in more than 384 seconds. For a sample of 71 patients, the mean time in which the medicine relieved pain was 386 seconds. Assume the population standard deviation is 23. Make the decision to reject or fail to reject the null hypothesis.

Solution Steps

The standard error \( SE \) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{23}{\sqrt{71}} \approx 2.7296 \]

The test statistic \( Z \) is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{386 - 384}{2.7296} \approx 0.7327 \]

For a right-tailed test, the P-value is calculated as:

\[ P = 1 - T(z) \approx 0.2319 \]

We compare the P-value to the significance level \( \alpha = 0.05 \):

• If \( P < \alpha \), we reject the null hypothesis.
• If \( P \geq \alpha \), we fail to reject the null hypothesis.

In this case:

\[ 0.2319 \geq 0.05 \]

Thus, we fail to reject the null hypothesis.

Final Answer