**Q3. Manually perform a *z*-test. (7 pts.)**

According to the CDC report, the mean life expectancy in the US population is currently (*µ*) 82.6 years with a standard deviation (*σ*) of 10. A researcher examined the age of death of 250 people recently recorded in several Arizona hospitals and calculated the mean to be 76.5 years old. He runs a two-tailed Z test with *α* = .05 to see if Arizona has a significantly different life expectancy compared to the US population. Because the researcher is not predicting a direction, the hypotheses should be non-directional.  *For this question, leave answers in their original decimal places.*

A. What are the null hypothesis and alternative hypothesis? Write each one in words and with symbols.

There is no significant difference in life expectancy between Arizona and US population.

H0: 82.6

Null Hypothesis: 82.6

H1: 82.6

H1: Alternative/Research Hypothesis: 82.6

B. Calculate the *standard error*. **(Show your calculation process.)**
**
*σ M* = *0.6325*

C. Calculate the Z statistic. **(Show your calculation process.)**

*z* = -9.64

D. Determine the critical *z* value(s). Explain how to find the answer.

Critical value is - 1.96, 1.96

E. What is the conclusion of the hypothesis test (do you “reject” or “fail to reject” the null hypothesis)? What is the rationale?

F. Answer the research question (you can use the wording from the hypotheses or explain it in another way).

G. Calculate the standardized effect size. No need to round. **(Show your calculation process.)**

*Cohen’s d =*

Let's go step-by-step through the process of performing the *z*-test based on the given data.

### **A. Hypotheses**

Since the researcher is conducting a two-tailed test, the hypotheses will be non-directional.

- **Null Hypothesis (H₀):** There is no significant difference in life expectancy between Arizona and the US population.
  - Symbolically: \( H_0: \mu = 82.6 \)
  
- **Alternative Hypothesis (H₁):** There is a significant difference in life expectancy between Arizona and the US population.
  - Symbolically: \( H_1: \mu \neq 82.6 \)

### **B. Calculate the Standard Error**

The standard error of the mean (SE or \(\sigma_M\)) is calculated using the formula:

\[
\sigma_M = \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\sigma = 10\) (the population standard deviation),
- \(n = 250\) (the sample size).

Now, calculate:

\[
\sigma_M = \frac{10}{\sqrt{250}} = \frac{10}{15.8114} \approx 0.6325
\]

Thus, the standard error \(\sigma_M = 0.6325\).

### **C. Calculate the Z Statistic**

The *z*-statistic is calculated using the formula:

\[
z = \frac{\bar{X} - \mu}{\sigma_M}
\]

Where:
- \(\bar{X} = 76.5\) (the sample mean),
- \(\mu = 82.6\) (the population mean),
- \(\sigma_M = 0.6325\) (the standard error).

Now, calculate:

\[
z = \frac{76.5 - 82.6}{0.6325} = \frac{-6.1}{0.6325} \approx -9.64
\]

Thus, the *z*-statistic \( z = -9.64 \).

### **D. Determine the Critical Z Values**

Since this is a two-tailed test with \(\alpha = 0.05\), we divide \(\alpha\) by 2 to find the critical values for each tail. For a two-tailed test with \(\alpha = 0.05\):

\[
\alpha/2 = 0.025
\]

Using the standard normal distribution, the critical values for a two-tailed test at the 0.05 significance level are:

\[
z_{critical} = \pm 1.96
\]

These critical values mark the rejection regions on both tails of the normal distribution.

### **E. Conclusion of the Hypothesis Test**

The computed *z*-statistic is \( z = -9.64 \), which is far outside the critical range of \(-1.96\) to \(1.96\). Since \( z = -9.64 \) lies in the rejection region, we reject the null hypothesis.

- **Conclusion:** We reject the null hypothesis, meaning there is significant evidence that the life expectancy in Arizona is different from that of the US population.

### **F. Answer the Research Question**

Based on the results of the hypothesis test, we can conclude that the life expectancy in Arizona hospitals (76.5 years) is significantly different from the national average of 82.6 years. Therefore, Arizona's life expectancy is lower than the US population's life expectancy.

### **G. Calculate the Standardized Effect Size (Cohen's d)**

Cohen's *d* is calculated using the formula:

\[
d = \frac{\bar{X} - \mu}{\sigma}
\]

Where:
- \(\bar{X} = 76.5\),
- \(\mu = 82.6\),
- \(\sigma = 10\).

Now, calculate:

\[
d = \frac{76.5 - 82.6}{10} = \frac{-6.1}{10} = -0.61
\]

Thus, Cohen’s *d* = \(-0.61\), which indicates a medium to large effect size (in absolute value).

---

Would you like more details or clarifications on any part?

### Related Questions:
1. How does a one-tailed test differ from a two-tailed test in hypothesis testing?
2. What is the interpretation of Cohen’s *d* in terms of effect size?
3. How do sample size and population standard deviation affect the standard error?
4. What does rejecting the null hypothesis imply in the context of real-world research?
5. How would you adjust the test if you wanted to predict that Arizona’s life expectancy is lower?

### Tip:
In hypothesis testing, always pay attention to the type of test (one-tailed vs. two-tailed) as it affects the critical values and the interpretation of the results.

Q3. Manually perform a *z*-test. (7 pts.) According to the CDC report, the mean life expectancy in the US population is currently (*µ*) 82.6 years with a standard deviation (*σ*) of 10. A researcher examined the age of death of 250 people recently recorded in several Arizona hospitals and calculated the mean to be 76.5 years old. He runs a two-tailed Z test with *α* = .05 to see if Arizona has a significantly different life expectancy compared to the US population. Because the researcher is not predicting a direction, the hypotheses should be non-directional. For this question, leave answers in their original decimal places.

Learn how to manually perform a z-test to determine if there is a significant difference in life expectancy between Arizona and the US population. This problem involves calculating standard error, z-statistic, and critical values, suitable for undergraduate statistics courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation