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.