he National Health Statistics Reports described a study in which a sample of  342 one-year-old baby boys were weighed. Their mean weight was 24.4 pounds with standard deviation 5.3  pounds. A pediatrician claims that the mean weight of one-year-old boys is less than 25  pounds. Do the data provide convincing evidence that the pediatrician's claim is true? Use the a=.1  level of significance and the p-value method with the TI-84 Plus calculator.

To determine if the pediatrician's claim that the mean weight of one-year-old boys is less than 25 pounds is supported by the data, we will perform a hypothesis test using the sample data provided. We will use the p-value method with a significance level of \(\alpha = 0.1\).

Here are the steps for performing the hypothesis test:

### Step 1: State the Hypotheses
- Null Hypothesis (\(H_0\)): \(\mu \geq 25\) (The mean weight is 25 pounds or more)
- Alternative Hypothesis (\(H_a\)): \(\mu < 25\) (The mean weight is less than 25 pounds)

### Step 2: Collect the Sample Data
- Sample mean (\(\bar{x}\)) = 24.4 pounds
- Sample standard deviation (\(s\)) = 5.3 pounds
- Sample size (\(n\)) = 342

### Step 3: Calculate the Test Statistic
The test statistic for a one-sample t-test is calculated using the formula:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis,
- \(s\) is the sample standard deviation,
- \(n\) is the sample size.

Let's calculate the test statistic:

\[
t = \frac{24.4 - 25}{5.3 / \sqrt{342}} \approx \frac{-0.6}{0.2868} \approx -2.092
\]

### Step 4: Determine the p-value
Using the TI-84 Plus calculator, we will find the p-value associated with the test statistic \( t \).

1. Press `2nd` then `VARS` to access the DISTR menu.
2. Select `tcdf(`.
3. Enter the lower bound (since we are interested in the left tail, it will be a very large negative number, such as -9999), the upper bound (our test statistic, which is -2.092), and the degrees of freedom (which is \( n - 1 = 342 - 1 = 341 \)).

The input will look like this: `tcdf(-9999, -2.092, 341)`

The calculator will return the p-value.

### Step 5: Compare the p-value with the Significance Level
- If the p-value is less than \(\alpha = 0.1\), we reject the null hypothesis.
- If the p-value is greater than or equal to \(\alpha = 0.1\), we fail to reject the null hypothesis.

### Step 6: Conclusion
Let's find the p-value using the TI-84 Plus:

Assume the p-value calculated by the calculator is approximately 0.018 (this is an approximate value for illustrative purposes; please use the TI-84 Plus to get the exact value).

Since \( p \approx 0.018 \) is less than \(\alpha = 0.1\), we reject the null hypothesis.

### Conclusion
There is convincing evidence at the \(\alpha = 0.1\) level of significance to support the pediatrician's claim that the mean weight of one-year-old boys is less than 25 pounds.

---

Do you want more details or have any questions? Here are some related questions to consider:

1. What is the purpose of the null hypothesis in hypothesis testing?
2. How do you interpret the p-value in the context of hypothesis testing?
3. Why is it important to define the significance level (\(\alpha\)) before conducting the test?
4. What assumptions must be met to use the t-test for hypothesis testing?
5. How would the results change if we used a significance level of \(\alpha = 0.05\)?
6. Can you explain the difference between a one-tailed and a two-tailed test?
7. How does sample size affect the outcome of a hypothesis test?
8. What are Type I and Type II errors in hypothesis testing?

**Tip:** Always verify the assumptions of the test you are using, such as normality and independence of the sample data, to ensure the validity of your results.

Explore whether the data from a sample of 342 one-year-old baby boys support a pediatrician's claim that their mean weight is less than 25 pounds. Includes step-by-step hypothesis testing using the p-value method with the TI-84 Plus calculator.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation