Fall 2024 Chapter 11 Part 2: Hypothesis Testing

Welcome back to Professor Baker's Math Class! In this section, we'll continue our journey into the world of hypothesis testing, focusing on applying these concepts to various real-world problems. We'll be using the principles discussed earlier to analyze different scenarios and draw meaningful conclusions. Let's dive in!

Hypothesis Testing for Proportions

Let's start with an example involving proportions. Imagine the city of Savannah wants to gauge its residents' support for building a toll bridge. A survey of 420 residents reveals that 228 favor the bridge. The mayor wants to know if a majority supports the bridge before holding a referendum, using a significance level of $\alpha = 0.01$.

Here's how we can approach this problem:

  1. Define the hypotheses:
    • Null Hypothesis ($H_0$): $p = 0.50$ (The proportion of residents supporting the bridge is 50%)
    • Alternative Hypothesis ($H_a$): $p > 0.50$ (The proportion of residents supporting the bridge is greater than 50%)
  2. Calculate the sample proportion:

    $\hat{p} = \frac{228}{420} = 0.54$

  3. Calculate the standard error:

    $\sigma_{\hat{p}} = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.50(1-0.50)}{420}} = 0.024$

  4. Compute the test statistic (Z-score):

    $Z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} = \frac{0.54 - 0.50}{0.024} = 1.67$

  5. Determine the critical value:

    For $\alpha = 0.01$ and a right-tailed test, the critical value is approximately 2.326.

  6. Make a decision:

    Since the calculated Z-score (1.67) is less than the critical value (2.326), we fail to reject the null hypothesis. There isn't enough evidence to conclude that a majority of residents support the toll bridge at the 0.01 significance level.

Another Example: Airline Carry-On Luggage

A commercial airline observes that 68% of passengers typically carry on luggage. After selecting 300 passengers, they find 237 with carry-ons. They want to test if the proportion has increased, using $\alpha = 0.01$.

  1. Define the hypotheses:
    • $H_0: p = 0.68$
    • $H_a: p > 0.68$
  2. Calculate the sample proportion:

    $\hat{p} = \frac{237}{300} = 0.79$

  3. Calculate the standard error:

    $\sigma = \sqrt{\frac{0.68(1-0.68)}{300}} = 0.027$

  4. Compute the test statistic (Z-score):

    $Z = \frac{0.79 - 0.68}{0.027} = 4.07$

  5. Determine the critical value:

    For $\alpha = 0.01$ and a right-tailed test, the critical value is approximately 2.326.

  6. Make a decision:

    Since the calculated Z-score (4.07) is greater than the critical value (2.326), we reject the null hypothesis. There is overwhelming evidence to suggest that the proportion of passengers with carry-on luggage has increased.

Keep practicing these examples, and you'll become a master of hypothesis testing! Good luck, and see you in the next chapter!