5.8 Sampling Distributions for Differences in Sample Means

• The sampling distribution of the difference in sample means is always normal regardless of population or sample size

  False
• What does the sampling distribution of the difference in sample means refer to?

  Probability distribution of differences between sample means
• What is the mean of the sampling distribution of the difference in sample means?

  \(\mu_1 - \mu_2\)
• The form of the sampling distribution of the difference in sample means is always normal

  False
• Steps to calculate the sampling distribution of the difference in sample means:

  1️⃣ Calculate the difference in sample means \(\bar{x}_1 - \bar{x}_2\)

  2️⃣ Determine if the samples are independent

  3️⃣ Check if both populations are normal or sample sizes are large

  4️⃣ Calculate the mean: \(\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2\)

  5️⃣ Calculate the standard deviation: \(\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)
• The sampling distribution of the difference in sample means is the probability distribution of differences between sample means
• Independent samples are taken from two populations with a direct relationship

  False
• The standard deviation of the sampling distribution of the difference in sample means is calculated as \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}
• Match the concept with its description:

  Independent Samples ↔️ No influence between samples

  Dependent Samples ↔️ Observations are related

  Mean of Sampling Distribution ↔️ $\mu_{\bar{x}_{1} - \bar{x}_{2}} =$$\mu_{1} - \mu_{2}$

  Standard Deviation of Sampling Distribution ↔️ $\sigma_{\bar{x}_{1} - \bar{x}_{2}} =$ \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}
• What is another name for dependent samples?

  Paired samples
• The sampling distribution of the difference in sample means is always normal regardless of the population distributions.

  False
• How is the difference in sample means calculated?

  \(\bar{x}_1 - \bar{x}_2\)
• The sampling distribution of the difference in sample means is always normal

  False
• Match the property with its corresponding distribution:

  Large, entire population ↔️ Population Distribution

  Based on sample sizes $n_{1}$ and $n_{2}$ ↔️ Sampling Distribution of the Difference in Sample Means
• Match the property with its corresponding distribution:

  Large, entire population ↔️ Population Distribution

  Based on sample sizes \(n_1\) and \(n_2\) ↔️ Sampling Distribution of the Difference in Sample Means
• Steps to calculate the sampling distribution of the difference in sample means:

  1️⃣ Calculate the difference in sample means \(\bar{x}_1 - \bar{x}_2\)

  2️⃣ Determine if the samples are independent

  3️⃣ Check if both populations are normal or sample sizes are large

  4️⃣ Calculate the mean: \(\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2\)

  5️⃣ Calculate the standard deviation: \(\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)
• What does it mean for samples to be independent in the context of the sampling distribution of the difference in sample means?

  No relationship between samples
• The standard error of the difference in sample means represents the variability we can expect in the difference between sample means
• What equals the difference between population means in the sampling distribution of the difference in sample means?

  $\mu_{1} - \mu_{2}$
• The sampling distribution of the difference in sample means is generally normal if both populations are normal or both sample sizes are large (≥30).

  True
• The standard error of the difference in sample means measures the variability of the differences between sample means
• Match the type of sample with an example:

  Independent Samples ↔️ Comparing test scores between two different classes

  Dependent Samples ↔️ Comparing pre-test and post-test scores for the same group of students
• Dependent samples are often used for comparisons between groups.

  False
• The Z-test should be used when the population standard deviations are unknown.

  False
• The t-test assumes the sampling distribution of the difference in sample means follows a t-distribution.


  True
• Independent samples are taken from two different populations with no relationship between
• The standard deviation of the sampling distribution of the difference in sample means is calculated as \(\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)
• What is the mean of the sampling distribution of the difference in sample means in terms of population means?

  \(\mu_1 - \mu_2\)
• What does it mean for samples to be independent in the context of the sampling distribution of the difference in sample means?

  No relationship between them
• How is the standard deviation of the sampling distribution of the difference in sample means calculated?

  \(\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)
• What is the formula for calculating the difference in sample means?

  \(\bar{x}_1 - \bar{x}_2\)
• How is the standard deviation of the sampling distribution of the difference in sample means calculated?

  \(\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)
• The sampling distribution of the difference in sample means is approximately normal under certain conditions.


  True
• What is the standard error of the difference in sample means denoted as?

  $\sigma_{\bar{x}_{1} - \bar{x}_{2}}$
• Steps to understand the sampling distribution of the difference in sample means:

  1️⃣ Recognize independent samples

  2️⃣ Calculate the difference in sample means

  3️⃣ Determine the shape (approximately normal)

  4️⃣ Find the mean: $\mu_{\bar{x}_{1} - \bar{x}_{2}} =$$\mu_{1} - \mu_{2}$

  5️⃣ Calculate the standard deviation: $\sigma_{\bar{x}_{1} - \bar{x}_{2}} =$ \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}
• The sampling distribution of the difference in sample means is the probability distribution of differences between sample means
• What is the statistic of interest in the sampling distribution of the difference in sample means?

  $\bar{x}_{1} - \bar{x}_{2}$
• Dependent samples, also known as paired samples, involve observations that are related
• What is the relationship between observations in independent samples?

  Unrelated
• Steps for hypothesis testing using the Z-test or t-test

  1️⃣ State the null and alternative hypotheses

  2️⃣ Determine the appropriate test statistic (Z or t)

  3️⃣ Calculate the test statistic and the p-value

  4️⃣ Compare the p-value to the significance level (α)

  5️⃣ Make a conclusion about the null hypothesis