Week 7: Single Population Hypothesis Testing Notes

Learning Outcomes and Course Objectives

• LO1: Develop an understanding of the concepts of point estimation.

• LO2: Understand the methodology used to estimate the population mean ($\mu$).

• LO3: Understand the methodology used to estimate the population proportion ($p$).

• LO4: Understand the fundamental basics of hypothesis testing.

Overview of Weekly Content

• Application of the principles of sampling distributions to estimate population parameters.

• Understanding the basic steps involved in Hypothesis Testing.

• Specific application of hypothesis testing for a population proportion ($p$).

• Specific application of hypothesis testing for a population mean ($\mu$).

Case Study: Regional vs. Urban Migration and Wages

This week uses a specific researcher case study to illustrate hypothesis testing principles:

• Background: A researcher finds that approximately $23.4\%$ of working people previously lived in regional Australia. It is believed this proportion has increased following the COVID-19 pandemic.

• Claims: There are claims that pandemic disruptions and restrictions were greatest in capital cities, while job growth was strongest in other (regional) areas.

• National Data: The national annual wage for 2023 is estimated to be $\text{AU\$98,176}$.

• Population of Interest: All working people in Australia.

• Sample Characteristics: To test these claims, the researcher collects a random sample of $n = 1,000$ individuals working in 2023.

• Variables Recorded:

  • Region: Coded as $1$ if the individual lives in regional Australia and $0$ if they live in an urban area.

  • Wage: Annual wage recorded in dollars.

Summary Statistics and Data Description

The following table summarizes the data collected from the sample of $n = 1,000$:

From Sampling Distribution to Estimation

Sample statistics serve as reliable and consistent estimators for population parameters:

• Sample Mean ( $\bar{X}$ ):

  • It is an unbiased estimator: $E(\bar{X}) = \mu$.

  • Variance: $\text{var}(\bar{X}) = \frac{\sigma^2}{n}$.

  • Consistency: The variance converges toward zero as the sample size ($n$) grows.

• Sample Proportion ( $\hat{p}$ ):

  • It is an unbiased estimator: $E(\hat{p}) = p$.

  • Variance: $\text{var}(\hat{p}) = \frac{p(1-p)}{n}$.

  • Consistency: The variance converges toward zero as the sample size ($n$) grows.

Steps of Hypothesis Testing

1. State the Hypotheses: Define the Null ($H_0$) and Alternative ($H_A$) hypotheses about a population parameter based on the research claim (not the sample statistic).

2. Specify the Decision Rule: Define the criteria for rejecting the null hypothesis, including the significance level ($\alpha$) and the critical value or p-value.

3. Calculate the Test Statistic: Compare the sample estimate with the hypothesized value. Identify the appropriate distribution (e.g., $t, n - 1$).

4. Apply the Decision Rule: Based on the evidence and the threshold, decide to either reject or not reject the null hypothesis.

5. Make the Decision and Draw Conclusions: Interpret the statistical result within the context of the real-world problem.

Hypothesis Testing: Population Proportions

Step 1: Set up the Hypothesis

• Null Hypothesis ($H_0$): The statement assumed true. For proportion $p$, it is typically set as equality to a value.

  • $H_0: p = 0.234$ (The proportion is the same as the pre-pandemic level).

• Alternative Hypothesis ($H_A$): The statement considered if $H_0$ is rejected.

  • Upper-tail Test: H_A: p > 0.234 (Used to test if the proportion has increased).

  • Lower-tail Test: H_A: p < 0.234 (Used to test if the proportion has decreased).

  • Two-tail Test: $H_A: p \neq 0.234$ (Used to test if the proportion is different).

Step 2: Decision Rule (Critical Value Approach)

• Significance Level ($\alpha$): A predetermined threshold (e.g., $0.01, 0.05, 0.10$).

• Example calculation: For $\alpha = 0.05$ and $df = n - 1 = 999$, the critical value using Excel is t.INV(0.95, 999) = 1.645.

• Decision Rule: Reject $H_0$ if the test statistic > 1.645.

Step 3: Calculate the Test Statistic

• The test statistic for a population proportion is: $t = \frac{\hat{p} - p_0}{\sqrt{\frac{1}{n}\hat{p}(1 - \hat{p})}}$

• Where:

  • $\hat{p}$ is the sample proportion ($\frac{270}{1000} = 0.270$).

  • $p_0$ is the hypothesized proportion ($0.234$).

  • The standard error is $\text{se}(\hat{p}) = \sqrt{\frac{1}{1000}0.270(1 - 0.270)} = 0.0140$.

• Calculation: $t = \frac{0.270 - 0.234}{0.0140} = 2.564$

Step 4 & 5: Decision and Conclusion

• Since 2.564 > 1.645, the test statistic falls in the rejection region.

• Decision: Reject $H_0$.

• Conclusion: At a $5\%$ significance level, the sample provides sufficient evidence that the proportion of the population living in regional areas has increased from the pre-covid proportion of $23.4\%$.

Hypothesis Testing Using the P-value

• P-value Definition: If the null hypothesis is true, the p-value is the probability of observing a statistic as extreme as the sample proportion obtained.

• Decision Rule: Reject $H_0$ if p\text{-value} < \alpha.

• Calculation (Upper-tail): p\text{-value} = P(t_{n-1} > t).

• Example Calculation: Using the previous proportion test ($t = 2.564$ and $df = 999$):

  • Excel: = 1 - T.DIST(2.564, 999, TRUE) = 0.0052.

• Since 0.0052 < 0.05, the decision is to Reject $H_0$.

Hypothesis Testing: Population Mean (Two-tailed Test)

Scenario: Is the average wage in urban areas different from $\text{\$100,000}$?

• Hypotheses:

  • $H_0: \mu_2 = 100,000$

  • $H_A: \mu_2 \neq 100,000$

• Parameters: $E(X_2) = \mu_2$, $\text{Var}(X_2) = \sigma_2^2$.

Test Statistic and Distribution

• Formula: $t = \frac{\bar{X}_2 - \mu_0}{\text{se}(\bar{X}_2)}$ where $\text{se}(\bar{X}_2) = \frac{s}{\sqrt{n_2}}$.

• Sample data for urban workers: $n_2 = 730$, $\bar{X}_2 = 100,438.3$, $s = 26,198.80$.

• Calculation:

  • $\text{se}(\bar{X}_2) = \frac{26,198.80}{\sqrt{730}} = 969.6611$

  • $t = \frac{100,438.3 - 100,000}{969.6611} = 0.4520$

• Distribution: follows $t(n-1) = t(729)$.

Decision Rule and Decision

• Critical Value Approach: For $\alpha = 0.05$, the critical values are $\pm 1.963$. Reject if |t| > 1.963.

• P-value Approach: p\text{-value} = 2 \times P(t_{729} > 0.452) = 0.6514 (calculated via Excel T.DIST.2T(ABS(0.452), 729)).

• Result: 0.4520 < 1.963, and 0.6514 > 0.05.

• Decision: Do Not Reject $H_0$.

• Conclusion: At a $5\%$ significance level, there is insufficient evidence to conclude the average wage in urban areas is different from $\text{\$100,000}$.

Hypothesis Testing: Population Mean (Lower-tail Test)

Scenario: Is the average wage in regional areas less than $\text{\$100,000}$?

• Hypotheses:

  • $H_0: \mu_1 = 100,000$

  • H_A: \mu_1 < 100,000

• Parameters: $n_1 = 270$, $\bar{X}_1 = 90,927.84$, $s_1 = 22,736.92$.

Test Statistic calculation

• $t = \frac{90,927.84 - 100,000}{\frac{22,736.92}{\sqrt{270}}} = -6.55$

Decision Rule and Decision

• Critical Value: For $\alpha = 0.05$ and $df = 269$, Critical Value = -1.6505 (using Excel t.INV(0.05, 269)).

• Rule: Reject $H_0$ if t < -1.6505.

• Decision: Since -6.55 < -1.6505, Reject $H_0$.

• P-value: P(t_{269} < -6.55) \approx 0.0000, which is less than $0.05$.

• Conclusion: At a $5\%$ significance level, there is sufficient evidence that the mean wage of working people in regional areas is less than $\text{\$100,000}$.