Let's break down the solution step-by-step to understand the hypothesis test for the mean Body Mass Index (BMI) based on the provided data.
### Step 1: Formulate the Null and Alternative Hypotheses
From the data, we need to determine whether the mean BMI is greater than 25. This establishes our null and alternative hypotheses.
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean BMI is equal to 25.
- Alternative Hypothesis ([tex]\( H_a \)[/tex]): The mean BMI is greater than 25.
In symbolic form:
- [tex]\( H_0 \)[/tex]: [tex]\( \mu = 25 \)[/tex]
- [tex]\( H_a \)[/tex]: [tex]\( \mu > 25 \)[/tex]
So, the correct answer for the hypotheses is:
C. [tex]\( H_0: \mu=25 \)[/tex]; [tex]\( H_a: \mu>25 \)[/tex]
### Step 2: Identify the Test Statistic
The test statistic helps to determine if the sample mean is significantly different from the hypothesized population mean. Here, the test statistic provided is [tex]\( T \)[/tex]:
[tex]\[
T = 2.49
\][/tex]
So, the test statistic is:
[tex]\[\boxed{2.49}\][/tex]
### Step 3: Determine the p-value
The p-value indicates the probability of obtaining the observed sample results if the null hypothesis is true. The p-value given in this problem is:
[tex]\[
\text{p-value} = 0.008
\][/tex]
So, the p-value is:
[tex]\[\boxed{0.008}\][/tex]
### Step 4: Compare the p-value with the Significance Level
The significance level ([tex]\( \alpha \)[/tex]) is 0.05. To determine whether to reject the null hypothesis, compare the p-value with [tex]\( \alpha \)[/tex]:
- If p-value [tex]\( \leq \alpha \)[/tex], reject [tex]\( H_0 \)[/tex].
- If p-value [tex]\( > \alpha \)[/tex], do not reject [tex]\( H_0 \)[/tex].
In this case:
[tex]\[
0.008 \leq 0.05
\][/tex]
Since the p-value (0.008) is less than the significance level (0.05), we reject the null hypothesis.
### Conclusion
Based on the hypothesis test, we have sufficient evidence to conclude that the mean BMI is greater than 25 at the 0.05 significance level.
### Summary of Answers:
1. The correct hypotheses are:
C. [tex]\( H_0: \mu=25 \)[/tex]; [tex]\( H_a: \mu>25 \)[/tex]
2. The test statistic is:
[tex]\[\boxed{2.49}\][/tex]
3. The p-value is:
[tex]\[\boxed{0.008}\][/tex]
