Let's analyze the given problem step-by-step:
### Step 1: Defining the hypotheses
We are given the following hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): [tex]\(\mu = 245\)[/tex]
- Alternative Hypothesis ([tex]$H_a$[/tex]): [tex]\(\mu \neq 245\)[/tex]
### Step 2: Given data and parameters
- Sample mean ([tex]\(\bar{x}\)[/tex]): 250 ppm
- Population mean ([tex]\(\mu\)[/tex]): 245 ppm (under the null hypothesis)
- Sample standard deviation ([tex]\(s\)[/tex]): 12 ppm
- Sample size ([tex]\(n\)[/tex]): 64 pills
- Significance level ([tex]\(\alpha\)[/tex]): 0.01
### Step 3: Compute the test statistic
We are given the test statistic directly in the problem:
- T-test statistic: 3.33
### Step 4: P-value
We are also provided with the p-value:
- P-value: 0.0014
### Step 5: Compare the P-value with the significance level
To make a decision, we compare the p-value to the significance level ([tex]\(\alpha = 0.01\)[/tex]):
- P-value ([tex]\(0.0014\)[/tex]) < Significance level [tex]\(\alpha\)[/tex] ([tex]\(0.01\)[/tex])
### Step 6: Conclusion
Since the p-value (0.0014) is less than the significance level (0.01), we reject the null hypothesis [tex]\(H_0\)[/tex].
### Step 7: Final conclusion
Based on the comparison of the p-value to the significance level, we can conclude that there is sufficient evidence to reject the null hypothesis. Therefore, we conclude that the average amount of the active ingredient in the pills is significantly different from 245 ppm.
### Conclusion Summary
The correct conclusion is: "Reject the null hypothesis [tex]\(H_0: \mu = 245\)[/tex]."
In summary:
- Sample mean: 250 ppm
- Population mean (under [tex]\(H_0\)[/tex]): 245 ppm
- Sample standard deviation: 12 ppm
- Sample size: 64 pills
- T-test statistic: 3.33
- P-value: 0.0014
- Significance level: 0.01
This leads us to reject the null hypothesis and conclude that the actual average amount of the active ingredient in the allergy medicine pills is not 245 ppm.
