Big babies: The National Health Statistics Reports described a study in which a sample of  one-year-old baby boys were weighed. Their mean weight was  pounds with standard deviation  pounds. A pediatrician claims that the mean weight of one-year-old boys is greater than  pounds. Do the data provide convincing evidence that the pediatrician's claim is true? Use the  level of significance and the critical value method with the Critical Values for the Student's t Distribution Table.

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Part 1 of 5
(a) State the appropriate null and alternate hypotheses.
 
 
 
 
This hypothesis test is a  test.
Part: 1 / 51 of 5 Parts Complete
Part 2 of 5
Find the critical value(s). Round the answer(s) to three decimal places. If there is more than one critical value, separate them with commas.

Critical value(s): 


To solve this problem, we need to perform a one-sample \( t \)-test to see if there is sufficient evidence to support the pediatrician's claim that the mean weight of one-year-old boys is greater than a certain weight.

### Part 1: Setting Up Hypotheses

1. **Null Hypothesis** \( H_0 \): The mean weight of one-year-old boys is equal to the given weight (let’s denote it as \( \mu_0 \)).
   
   \[
   H_0: \mu = \mu_0
   \]

2. **Alternate Hypothesis** \( H_a \): The mean weight of one-year-old boys is greater than the given weight \( \mu_0 \).
   
   \[
   H_a: \mu > \mu_0
   \]

This hypothesis test is a **one-tailed (right-tailed) t-test**, as we're testing if the mean weight is **greater than** a specified value.

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### Part 2: Finding the Critical Value(s)

The critical value(s) depend on:
- The level of significance \( \alpha \),
- The degrees of freedom, which are calculated as \( \text{df} = n - 1 \), where \( n \) is the sample size.

To find the critical value(s) for this one-tailed test:
1. Identify the level of significance \( \alpha \).
2. Use the Student's \( t \)-distribution table to find the \( t \)-score that corresponds to a cumulative probability of \( 1 - \alpha \) for the given degrees of freedom.

Since some data details (such as \( \alpha \), sample size \( n \), and \( \mu_0 \)) are missing in this problem statement, please provide these values so that we can proceed accurately.

Let me know if you’d like additional details, or if there’s a specific part of this explanation you'd like to dive into further!

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### Relative Questions:
1. How do we determine the degrees of freedom in a \( t \)-test?
2. What is the difference between a one-tailed and a two-tailed \( t \)-test?
3. Why is the \( t \)-distribution used instead of the normal distribution in this scenario?
4. What effect does the significance level \( \alpha \) have on the critical value?
5. How would the hypothesis test differ if we were testing if the mean is simply different (not greater) than \( \mu_0 \)?

**Tip:** For a one-tailed test, always specify which direction (greater or less) aligns with the alternative hypothesis, as it determines your critical region.

Big babies: The National Health Statistics Reports described a study in which a sample of one-year-old baby boys were weighed. Their mean weight was pounds with standard deviation pounds. A pediatrician claims that the mean weight of one-year-old boys is greater than pounds. Do the data provide convincing evidence that the pediatrician's claim is true? Use the level of significance and the critical value method with the Critical Values for the Student's t Distribution Table.

Explore a hypothesis test for the mean weight of one-year-old boys using the one-sample t-test and critical value method. This example uses data from a sample study and tests the pediatrician's claim that the mean weight is greater than a specific value, applying a specified level of significance and critical values from the Student's t-Distribution table.

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