A researcher, studying stress is interested in the blood pressure measurements of chief executive officers CEOs of major corporations yes, good reason to believe that the main systolic blood pressure of CEOs of major corporations is different from 130 to MMHG which is the value reported in a possibly outdated journal article, he plans to perform a statistical test. He measures the systolic blood pressure of a random sample of CEOs of major corporations, and find the mean of the sample to be 126 MMHG and the standard deviation of the sample to be 15 MMHG.

What are the null hypothesis H0 and the alternative hypothesis H1 that should be used for the test? Suppose that the researcher decides not to reject the null hypothesis what sort of error might he be making? Suppose the true mean systolic blood pressure of CEOs of major corporations is 122MMHG fill in the blanks to describe a type two error.. “

Let's break down the components of the hypothesis test and the possible errors:

### Null Hypothesis (\(H_0\)):
The null hypothesis is the statement that there is no difference between the mean systolic blood pressure of CEOs and the value reported in the outdated article (130 mmHg). Formally, it is stated as:
\[ H_0: \mu = 130 \text{ mmHg} \]

### Alternative Hypothesis (\(H_1\) or \(H_a\)):
The alternative hypothesis is the researcher's claim that the mean systolic blood pressure of CEOs is different from 130 mmHg. Formally, it is stated as:
\[ H_1: \mu \neq 130 \text{ mmHg} \]

This is a two-tailed test because the researcher is interested in detecting any difference, whether higher or lower than 130 mmHg.

### Type II Error (\(\beta\)):
A Type II error occurs when the null hypothesis is not rejected, even though it is actually false. In other words, the researcher concludes that there is no significant difference between the true mean systolic blood pressure and 130 mmHg when, in reality, there is a difference.

### Filling in the Blanks for a Type II Error:
Given that the true mean systolic blood pressure of CEOs is 122 mmHg, a Type II error would occur if the researcher decides **not to reject the null hypothesis**. The sentence describing a Type II error can be filled in as follows:

*"A Type II error would occur if the researcher concludes that the mean systolic blood pressure of CEOs is not significantly different from 130 mmHg when, in fact, the true mean systolic blood pressure is 122 mmHg."*

### Summary:
- **Null Hypothesis (\(H_0\))**: The mean systolic blood pressure of CEOs is 130 mmHg (\(\mu = 130\)).
- **Alternative Hypothesis (\(H_1\))**: The mean systolic blood pressure of CEOs is not 130 mmHg (\(\mu \neq 130\)).
- **Type II Error**: Occurs if the researcher fails to reject \(H_0\) when the true mean is actually 122 mmHg.

Would you like to explore the statistical methods to calculate the probability of a Type II error, or do you have any other questions?

### Related Questions:
1. How do you calculate the power of a hypothesis test?
2. What factors influence the probability of making a Type II error?
3. How can the significance level (\(\alpha\)) affect the Type I and Type II error rates?
4. What sample size would minimize the risk of a Type II error in this test?
5. How do confidence intervals relate to hypothesis testing?
6. What is the difference between one-tailed and two-tailed tests?
7. How do you calculate the test statistic for this hypothesis test?
8. How would the results differ if the researcher used a one-tailed test instead?

### Tip:
To reduce the probability of making a Type II error, consider increasing the sample size or choosing a higher significance level (though this increases the chance of a Type I error). Balancing these risks is crucial in hypothesis testing.

Explore hypothesis testing for the mean systolic blood pressure of CEOs of major corporations. Understand the null and alternative hypotheses, type II error, and implications of statistical decisions. Example scenario with mean systolic blood pressure of 126 mmHg and standard deviation of 15 mmHg.

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