Objectives

Previously… (1/3)

The guiding principle of statistics is statistical thinking.

Statistical Thinking in the Data Science Life Cycle

Previously… (2/3)

Types of Inference

Previously… (3/3)

Types of Decision Errors

Parameter Estimation for One Proportion

Conditions

• Independent samples
• Success-failure outcomes is \(np \ge 10\) and \(n(1-p) \ge 10\).
• We can assume Normal approximation of the Binomial PMF.

Confidence Intervals for One Proportion

\[\hat{p} \pm z^{\star} SE_{\hat{p}}\]

• \(\hat{p}\) is the sample proportion
• \(z^{\star}\) is the critical z-score for a given confidence level
• \(SE_{\hat{p}} = \sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}\) is the standard error

Parameter Estimation for Two Proportions

Conditions

• Independent samples
• Success-failure outcomes is \(n_Ap_A \ge 10\), \(n_A(1-p_A) \ge 10\), \(n_Bp_B \ge 10\), and \(n_B(1-p_B) \ge 10\)
• We can assume Normal approximation of the Binomial PMF.

Confidence Intervals for Two Proportions

\[\hat{p}_B - \hat{p}_A \pm z^{\star} SE_{\hat{p}_B - \hat{p}_A}\]

• \(\hat{p}_A\) is the sample proportion for group \(A\)
• \(\hat{p}_B\) is the sample proportion for group \(B\)
• \(z^{\star}\) is the critical z-score for a given confidence level
• \(SE_{\hat{p}_B - \hat{p}_A} = \sqrt{ \frac{\hat{p}_A \left( 1-\hat{p}_A \right)}{n_A} + \frac{\hat{p}_B \left( 1-\hat{p}_B \right)}{n_B} }\) is the standard error

Hypothesis Testing for One Proportion (1/2)

Let \(p\) be the population parameter and \(p_0\) be the null value.

State Hypotheses

• Null Hypothesis \(H_0\): The population proportion remains unchanged. \[p = p_0\]
• Alternative Hypothesis \(H_A\): The population proportion has changed. \[p \ne p_0\]

Set Significance Value \(\alpha\)

Common values are \(\alpha = 0.10, 0.05, 0.01\). Note that \(\alpha\) is the Type I error rate.

Hypothesis Testing for One Proportion (2/2)

Compute the Test Statistic

\[z = \frac{\hat{p}-p_0}{SE_p}\]

• \(\hat{p}\) is the point estimate
• \(SE_p = \sqrt{\frac{p_0\left( 1-p_0 \right)}{n}}\) is the standard error of the null sampling distribution

Determine the P-Value

• You need to find \(P(Z \le z)\) for one-sided test or \(2 \times P(Z \le z)\) for a two sided test from \(Z \sim N(0,1)\), the standard normal distribution

Make a Decision and conclusion

• Reject \(H_0\) if the \(\text{p-value} < \alpha\): There is enough evidence to support \(H_A\).
• Fail to reject the \(H_0\) if the \(\text{p-value} \ge \alpha\): There is not enough evidence to support \(H_A\).

Hypothesis Testing for Two Proportions (1/2)

Let \(p_A\) and \(p_B\) be the population parameters for groups \(A\) and \(B\) respectively and \(p_0\) be the null value.

State Hypotheses

• Null Hypothesis \(H_0\): The difference in population proportion remains unchanged. \[p_B - p_A = p_0\]
• Alternative Hypothesis \(H_A\): The difference in population proportion has changed. \[p_B - p_A \ne p_0\]

Set Significance Value \(\alpha\)

Common values are \(\alpha = 0.10, 0.05, 0.01\). Note that \(\alpha\) is the Type I error rate.

Hypothesis Testing for Two Proportions (2/2)

Compute the Test Statistic

\[z = \frac{\left(\hat{p}_B-\hat{p}_A\right)-p_0}{SE_{p_B - p_A}}\]

• \(\hat{p}_A\) is the point estimate for group \(A\)
• \(\hat{p}_B\) is the point estimate for group \(B\)
• \(SE_{p_B - p_A} = \sqrt{\hat{p}_{pool}\left( 1-\hat{p}_{pool} \right)\left( \frac{1}{n_A} + \frac{1}{n_B} \right)}\) is the standard error
• \(\hat{p}_{pool} = \frac{n_A \hat{p}_A + n_B \hat{p}_B}{n_A + n_B}\)

Determine the P-Value

• You need to find \(P(Z \le z)\) for one-sided test or \(2 \times P(Z \le z)\) for a two sided test from \(Z \sim N(0,1)\), the standard normal distribution

Make a Decision and conclusion

• Reject \(H_0\) if the \(\text{p-value} < \alpha\): There is enough evidence to support \(H_A\).
• Fail to reject the \(H_0\) if the \(\text{p-value} \ge \alpha\): There is not enough evidence to support \(H_A\).

Activity: Rock-Paper-Scissors Hypothesis Testing

1. Make sure you have a copy of the W 3/19 Worksheet. This will be handed out physically and it is also digitally available on Moodle.
2. Work on your worksheet by yourself for 10 minutes. Please read the instructions carefully. Ask questions if anything need clarifications.
3. Get together with another student.
4. Discuss your results.
5. Submit your worksheet on Moodle as a .pdf file.

RPS Wins Data

Data collection was done on 3/19/2025

Number of Wins

RPS Proportion of Wins Data

Data collection was done on 3/19/2025

Proportions

Each group played 15 rounds for blind and 15 rounds for non-blind.

References