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A body mass index of more than 25 is considered unhealthful. The technology output given is from 50 randomly and independently selected people from a health agency's study. Test the hypothesis that the mean BMI is more than 25 using a significance level of 0.05. Assume that conditions are met.

One-Sample T Test of [tex]$\mu=25$[/tex] vs [tex]$\ \textgreater \ 25$[/tex]

\begin{tabular}{rrrrcc}
[tex]$N$[/tex] & Mean & StDev & SE Mean & T & P \\
50 & 27.421 & 6.888 & 0.974 & 2.49 & 0.008 \\
\hline
\end{tabular}

Determine the null and alternative hypotheses. Choose the correct answer below:

A. [tex]$H_0: \mu=25$[/tex]

B. [tex]$H_0: \mu \neq 25$[/tex]

C. [tex][tex]$H_0: \mu=25$[/tex][/tex] [tex]$H_a: \mu\ \textgreater \ 25$[/tex]

D. [tex]$H_0: \mu\ \textgreater \ 25$[/tex]

E. [tex][tex]$H_0: \mu=25$[/tex][/tex]

F. [tex]$H_0: \mu\ \textless \ 25$[/tex] [tex]$H_a: \mu \leq 25$[/tex] [tex][tex]$H_a: \mu\ \textless \ 25$[/tex][/tex] [tex]$H_a: \mu \geq 25$[/tex]

The test statistic is [tex]\square[/tex] (Type an integer or decimal rounded to two decimal places as needed.)

The p-value is [tex]\square[/tex] (Type an integer or decimal rounded to three decimal places as needed.)



Answer :

GinnyAnswer
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]