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.