A scientist has read that the mean birth weight of babies born at full term is 7.5 pounds. The scientist has good reason to believe that the mean birth weight of babies born at full term,  , is greater than this value and plans to perform a statistical test. She selects a random sample of birth weights of babies born at full term and finds the mean of the sample to be 7.7 pounds and the standard deviation to be 1.5 pounds.

Based on this information, complete the parts below. (a)What are the null hypothesis H0 and the alternative hypothesis H1 that should be used for the test? (b)Suppose that the scientist decides not to reject the null hypothesis. What sort of error might she be making? (c)Suppose the true mean birth weight of babies born at full term is  pounds. Fill in the blanks to describe a Type II error. A Type II error would be rejecting/falling to reject the hypothesis that is .....

### (a) Null Hypothesis \( H_0 \) and Alternative Hypothesis \( H_1 \):

1. **Null Hypothesis (H₀):** The mean birth weight of babies born at full term is equal to 7.5 pounds.
   - Mathematically: \( H_0: \mu = 7.5 \)

2. **Alternative Hypothesis (H₁):** The mean birth weight of babies born at full term is greater than 7.5 pounds.
   - Mathematically: \( H_1: \mu > 7.5 \)

This is a one-tailed (right-tailed) test because the scientist believes the mean birth weight is greater than 7.5 pounds.

### (b) Type of Error if the Scientist Does Not Reject the Null Hypothesis:

If the scientist decides not to reject the null hypothesis, she might be making a **Type II error**.

- A **Type II error** occurs when the null hypothesis is not rejected when it is actually false. In this case, the scientist fails to conclude that the mean birth weight is greater than 7.5 pounds when, in reality, it is.

### (c) Type II Error Description:

Suppose the true mean birth weight of babies born at full term is \( \mu > 7.5 \) pounds. 

A **Type II error** would be **failing to reject** the hypothesis that \( \mu = 7.5 \) when, in fact, the true mean is greater than 7.5 pounds.

To summarize:

- **Type I error:** Rejecting \( H_0 \) when \( H_0 \) is true.
- **Type II error:** Failing to reject \( H_0 \) when \( H_0 \) is false.

Do you want details on calculating the test statistic or other aspects of hypothesis testing?

### Related Questions:
1. What is the significance level (α) commonly used for hypothesis testing?
2. How do you calculate the test statistic in this case?
3. How would you interpret the p-value obtained from this test?
4. What assumptions must be satisfied to perform this hypothesis test?
5. How would you determine the power of the test?
6. What is the impact of sample size on Type I and Type II errors?
7. How do you calculate the probability of making a Type II error (β)?
8. What conclusions can be drawn if the null hypothesis is rejected?

#### Tip:
When designing a hypothesis test, it's important to balance the risks of Type I and Type II errors. Consider the consequences of each type of error in your specific context.

Explore hypothesis testing in statistics with a focus on mean birth weight analysis. Understand the concepts of null and alternative hypotheses, type I and type II errors, and the calculation of test statistics and p-values.

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