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I see the file contains several questions on data analysis, hypothesis testing, confidence intervals, and misconceptions about p-values. Below is a breakdown of the content:

1. **Confidence Interval Calculation (Q1)**: 
   You are tasked with calculating a 95% confidence interval for a sample mean (X̄ = 20) with a sample size of 25 and a known standard deviation (σ = 5). 
   - The confidence interval is calculated using the formula:
     \[
     CI = X̄ \pm Z_{\alpha/2} \left(\frac{\sigma}{\sqrt{N}}\right)
     \]
     Where:
     - \( Z_{\alpha/2} = 1.96 \) (for a 95% confidence level),
     - \( \sigma = 5 \),
     - \( N = 25 \).
     The bounds are approximately (18.04, 21.96).

2. **Hypothesis Testing for Team Scores (Q2)**: 
   Here, you calculated a sample mean of 80.89 based on the given data. Next, you are asked to compute a 95% confidence interval, determine if scores have improved since a new team member joined, and calculate the probability of the observed mean under the null hypothesis (H₀: μ = 78). This involves hypothesis testing with a one-tailed test.

3. **Z-Test on Life Expectancy (Q3)**: 
   You perform a Z-test comparing life expectancy in Arizona hospitals (mean = 76.5) against the US population (mean = 82.6, σ = 10, N = 250). The critical value for a two-tailed test at α = 0.05 is ±1.96. The Z-statistic is calculated as:
   \[
   Z = \frac{(X̄ - μ)}{\sigma / \sqrt{N}}
   \]
   Leading to a Z-score of -9.64, which suggests a strong deviation from the population mean.

4. **Misconceptions About Hypothesis Testing (Q4)**: 
   This section corrects common misconceptions about p-values and significance tests. For example, a p-value does not measure the probability that the null hypothesis is false, but instead, the probability of obtaining results as extreme as those observed, assuming the null hypothesis is true.

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Would you like further details on any specific calculations, explanations, or how to handle hypothesis tests?

Here are some questions that can deepen your understanding:
1. How would the confidence interval change if the sample size were larger?
2. What would the Z-value be for a different confidence level, say 99%?
3. How would you interpret the p-value obtained in Q2?
4. What effect does a larger sample size have on the hypothesis test in Q3?
5. What other misconceptions about p-values or significance testing can affect research conclusions?

**Tip**: When calculating confidence intervals or performing hypothesis testing, always ensure that the correct assumptions about normality, sample size, and standard deviation are met to ensure valid results.

Q1: Use the estimation method to calculate and interpret a confidence interval. Q2: Use the estimation method and compare models to test a hypothesis. Q3: Manually perform a z-test. Q4: Explain misconceptions about hypothesis testing.

This guide covers the calculation of confidence intervals, hypothesis testing, and Z-tests, including key concepts such as p-value interpretation and common misconceptions. It includes real-world examples of sample means, confidence intervals, and hypothesis testing in fields like team performance and life expectancy.

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