Questions: At the 5% significance level, test the researcher's claim assuming that the population standard deviation of pain-relieving times is 5 minutes. You may also assume that the sample is from a normal population. (Note: The average and the standard deviation of the data are respectively 72.34 minutes and 4.28 minutes.) Procedure: One mean Z Hypothesis Test ✓ Assumptions: (select everything that applies) - Normal population - Population standard deviation is unknown - Population standard deviation is known - Simple random sample - Sample size is greater than 30 - The number of positive and negative responses are both greater than 10 Step 1. Hypotheses Set-Up: Step 2. The significance level α =

At the 5% significance level, test the researcher's claim assuming that the population standard deviation of pain-relieving times is 5 minutes. You may also assume that the sample is from a normal population. (Note: The average and the standard deviation of the data are respectively 72.34 minutes and 4.28 minutes.)
Procedure: One mean Z Hypothesis Test ✓
Assumptions: (select everything that applies)
- Normal population
- Population standard deviation is unknown
- Population standard deviation is known
- Simple random sample
- Sample size is greater than 30
- The number of positive and negative responses are both greater than 10

Step 1. Hypotheses Set-Up:

Step 2. The significance level α =

Transcript text: At the $5 \%$ significance level, test the researcher's claim assuming that the population standard deviation of pain-relieving times is 5 minutes. You may also assume that the sample is from a normal population. (Note: The average and the standard deviation of the data are respectively 72.34 minutes and 4.28 minutes.) Procedure: $\square$ One mean Z Hypothesis Test $\checkmark$ $\square$ Assumptions: (select everything that applies) Normal population Population standard deviation is unknown Population standard deviation is known Simple random sample Sample size is greater than 30 The number of positive and negative responses are both greater than 10 Step 1. Hypotheses Set-Up: Step 2. The significance level $\alpha=$ $\square$

Solution

Solution Steps

Step 1: Hypotheses Set-Up

We set up the hypotheses for the test as follows:

Null Hypothesis ($H_0$): $ \mu = 70 $
Alternative Hypothesis ($H_a$): $ \mu \neq 70 $

Step 2: Significance Level

The significance level is given as $ \alpha = 0.05 $.

Step 3: Calculate Standard Error

The standard error ($SE$) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{4.28}{\sqrt{28}} \approx 0.9449 \]

Step 4: Calculate Test Statistic

The test statistic ($Z$) is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{72.34 - 70}{0.9449} \approx 2.4764 \]

Step 5: Calculate P-value

For a two-tailed test, the p-value is calculated as: \[ P = 2 \times (1 - T(|z|)) \approx 0.0133 \]

Step 6: Decision Rule

Since the p-value $0.0133$ is less than the significance level $0.05$, we reject the null hypothesis.

Final Answer

The test statistic is $Z \approx 2.4764$ and the p-value is $P \approx 0.0133$. Therefore, we reject the null hypothesis in favor of the alternative hypothesis.

$\boxed{H_a \text{ is supported, indicating that the mean pain-relieving time is significantly different from } 70.}$

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