Activity 24

MATH 216: Statistical Thinking

Activity 1: Quiz

  1. Given the following sample data: \(\bar{x} = 80\), \(s = 6\), \(n = 25\), and \(\mu_0 = 75\), calculate the test statistic (\(z\)) for a one-tailed test where \(H_a: \mu > 75\).

  2. Use the formula:

    \[ z = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \]

  3. Discuss: After calculating your answer, discuss with a neighbor how you arrived at your result.

  • What is the critical value for \(\alpha = 0.05\)?
  • Would you reject the null hypothesis based on your calculated \(z\)-score?

Activity 2: Case Studies and Problem-Solving

  1. Case Study: A nutritionist claims that a new diet reduces the average calorie intake below 2200 calories/day. The sample data shows \(\bar{x} = 2150\), \(s = 120\), \(n = 35\).
  2. Task: Conduct a hypothesis test to determine if the data supports the nutritionist’s claim at \(\alpha = 0.05\).
  3. Analysis: Calculate the test statistic (\(z\)) and p-value. Determine whether to reject the null hypothesis.
  4. Discussion: Discuss the implications of your findings. What are the potential consequences of a Type I or Type II error in this context?

Activity 3: Group Activities with Real Data

An analyst claims that a new billing system changes the average monthly bill from $100.

  • Sample Data: \(\bar{x} = 105\), \(s = 10\), \(n = 40\)
  • Test: \(H_0: \mu = 100\) vs. \(H_a: \mu \neq 100\)
  • Calculation: \(z = \frac{105 - 100}{10/\sqrt{40}} \approx 3.16\)
  • Conclusion: Reject \(H_0\) if \(|z| > 1.96\) (at \(\alpha = 0.05\)).
  1. Based on the test statistic \(z = 2.53\), calculate the p-value and determine whether to reject the null hypothesis at a significance level of \(\alpha = 0.05\). Does the p-value suggest strong evidence against the null hypothesis?
  1. Construct the 95% confidence interval for the population mean using the sample data (\(\bar{x} = 105\), \(s = 10\), \(n = 40\)). Based on the confidence interval, would you reject the null hypothesis that the population mean is \(100\)?
  1. Based on the confidence interval, determine whether you would reject the null hypothesis that the population mean is \(100\).

Activity 4

The sample of 100 drug-injected rats yielded the results (in seconds) shown in the following Table. Conduct a hypothesis test to determine if the mean reaction time differs from 1.2 seconds at \(\alpha = 0.01\). Assume \(\bar{x} = 1.05\), \(s = 0.49\), \(n = 100\)

\[\begin{align*} H_0: \mu=1.2 \\ H_a: \mu \neq 1.2 \end{align*}\]

  • What is the calculated \(z\)-score?
  • What is the p-value, and does it support rejecting the null hypothesis?