6.10 Setting Up a Test for the Difference of Two Population Proportions

• The null hypothesis for testing the difference between two population proportions is written as $H_{0}: p_{1} =$$p_{2}$
• What is the independence condition that must be verified for conducting a two-sample proportion test?

  Independent samples
• In the pooled sample proportion formula, $x_{1}$ and $x_{2}$ represent the number of successes in the two samples.


  True
• In the pooled sample proportion formula, $x_{1}$ and $x_{2}$ represent the number of successes in the two samples.


  True
• The standard error for the difference of two population proportions measures the variability in the sampling distribution of $\hat{p}_{1} - \hat{p}_{2}$.


  True
• The null hypothesis for testing the difference between two population proportions assumes there is no difference
• What must be ensured for each population to satisfy the success-failure condition?

  Both $x$ and $n - x$ are $\geq 5$
• What is the pooled sample proportion if sample 1 has 35 successes out of 100 observations and sample 2 has 40 successes out of 120 observations?

  $\hat{p} \approx 0.341$
• What does $x_{1}$ represent in the formula for pooled sample proportion?


  Number of successes in sample 1
• The pooled sample proportion combines data from multiple samples to provide a single estimate of the proportion in the overall population.


  True
• If sample 1 has 35 successes out of 100 observations and sample 2 has 40 successes out of 120 observations, the pooled sample proportion is approximately 0.341
• What does $\hat{p}$ represent in the standard error formula?


  Pooled sample proportion
• In the z-test statistic formula, $\hat{p}_{1}$ and $\hat{p}_{2}$ are the sample proportions
• For a two-tailed test, the p-value is calculated as 2 times the probability of observing a z-statistic at least as extreme as the calculated value.
• If the p-value is less than or equal to the significance level, we reject the null hypothesis.
• Order the types of alternative hypotheses for a test of two population proportions based on the direction of the difference.

  1️⃣ Two-tailed: $H_{a}: p_{1} \neq p_{2}$

  2️⃣ Right-tailed: $H_{a}: p_{1} > p_{2}$

  3️⃣ Left-tailed: $H_{a}: p_{1} < p_{2}$
• For the success-failure condition, the number of successes and failures in each sample must both be at least 5
• What is the pooled sample proportion denoted as?

  $\hat{p}$
• What do $n_{1}$ and $n_{2}$ represent in the pooled sample proportion formula?


  Sample sizes
• Match the type of standard error with its formula:

  For the Difference of Two Population Proportions ↔️ \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_{1}} + \frac{1}{n_{2}}\right)}

  For One Population Proportion ↔️ $\sqrt{\frac{p(1 - p)}{n}}$

  For the Difference of Two Population Means ↔️ \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}
• What is the first condition that must be verified before conducting a test for the difference of two population proportions?

  Independent Samples
• The pooled sample proportion provides a single estimate of the proportion in the overall population
• Match the sample type with its proportion calculation:

  Sample 1 ↔️ $\frac{35}{100} =$$0.35$

  Sample 2 ↔️ $\frac{40}{120} \approx 0.33$

  Pooled Sample ↔️ $\frac{75}{220} \approx 0.341$
• The sample sizes $n_{1}$ and $n_{2}$ are used in the calculation of the pooled sample proportion.


  True
• What is the formula for calculating the pooled sample proportion?

  \hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}}</latex>
• The standard error for the difference of two population proportions measures the variability in the sampling distribution of the difference in sample proportions.


  True
• If \hat{p} = 0.4</latex>, $n_{1} =$$100$, and $n_{2} =$$150$, the standard error is approximately 0.06
• The pooled sample proportion is used in the denominator of the z-test statistic formula.


  True
• How is the p-value calculated for a right-tailed test?

  $P(Z \geq z)$
• What action do we take if the p-value is 0.02 and the significance level is 0.05?

  Reject the null hypothesis
• What does the null hypothesis assume in a test for the difference between two population proportions?

  No difference exists
• In a two-tailed alternative hypothesis, the null hypothesis is $H_{0}: p_{1} =$$p_{2}$ and the alternative hypothesis is $H_{a}: p_{1} \neq p_{2}$.


  True
• What is the formula to calculate the pooled sample proportion $\hat{p}$?


  \hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}}</latex>
• The formula for calculating the pooled sample proportion is \hat{p}
• The pooled sample proportion provides a single estimate of the population proportions
• Order the types of alternative hypotheses from most general to most specific:

  1️⃣ Two-tailed: H_{a}: p_{1} \neq p_{2}</latex>

  2️⃣ Right-tailed: $H_{a}: p_{1} > p_{2}$

  3️⃣ Left-tailed: $H_{a}: p_{1} < p_{2}$
• Both sample sizes ($n_{1}$ and $n_{2}$) must be at least 30 to ensure the sampling distributions follow a normal distribution.


  True
• The formula for the pooled sample proportion is $\hat{p} =$ \frac{x_{1} + x_{2}}{n_{1} + n_{2}}.


  True
• The formula for calculating the pooled sample proportion is \hat{p}
• If sample 1 has 35 successes out of 100 observations and sample 2 has 40 successes out of 120 observations, the pooled sample proportion is approximately 0.341