7.7 Confidence Intervals for the Difference of Two Means

• A confidence interval is likely to contain the true population parameter
• If the p-value is less than the significance level, you reject the null hypothesis.
• Steps to determine if the difference of two means is significant

  1️⃣ Calculate the difference of the means

  2️⃣ Compare to the null hypothesis

  3️⃣ Calculate the p-value

  4️⃣ Compare the p-value to the significance level

  5️⃣ Reject or fail to reject the null hypothesis
• What is a common significance level used in hypothesis testing?

  0.05
• Under what condition is the pooled t-test used?

  Equal variances
• The unpooled t-test is used when variances are assumed to be equal.

  False
• When should the pooled t-test be used?

  Variances are equal
• What condition related to data distribution must be met for both t-tests?

  Normality
• What does statistical significance indicate about observed differences?

  Probability they occurred by chance
• Wider confidence intervals imply less precision.
• What is the purpose of a confidence interval?

  Estimate the true parameter
• What is the null hypothesis when testing the difference of two means?

  $\bar{x}_{1} - \bar{x}_{2} =$$0$
• What is the formula for calculating the pooled variance?

  $s_{p}^{2} =$ \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2}
• The pooled t-test assumes that the variances of the two populations are equal
• What is the primary assumption that distinguishes a pooled t-test from an unpooled t-test?

  Equality of variances
• Steps in calculating a confidence interval for the difference of two means using an unpooled t-test

  1️⃣ Check assumptions

  2️⃣ Calculate sample statistics

  3️⃣ Determine degrees of freedom

  4️⃣ Find t-value

  5️⃣ Calculate confidence interval

  6️⃣ Interpret the interval
• What is a p-value in statistical significance testing?

  Probability of observing difference
• What does the null hypothesis claim about population means?

  No difference between means
• A smaller p-value indicates stronger evidence against the null
• The probability that the observed difference occurred by chance is called the significance
• The formula for the difference of two means is $\bar{x}_{1} - \bar{x}_{2}$
• What decision is made if the p-value is greater than or equal to the significance level?

  Fail to reject the null hypothesis
• What does the pooled variance formula $s_{p}^{2} =$ \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2} calculate?


  The combined variance of two samples
• The normality assumption for t-tests can be satisfied by large sample sizes due to the Central Limit Theorem.


  True
• The assumption of normality requires that the sample data is approximately normally distributed or the sample sizes are large enough to apply the Central Limit Theorem.
• The significance of a statistical test is the probability that the observed difference occurred by chance.


  True
• Match the aspect with its description:

  Confidence Interval ↔️ Range likely to contain the true population parameter

  Significance ↔️ Probability the observed difference occurred by chance
• The p-value is the probability that the observed difference could have occurred by chance if the null hypothesis is true.
• If the p-value is less than the significance level, you reject the null hypothesis.

  True
• The pooled t-test uses the pooled variance denoted by $s_{p}^{2}$, which is calculated using the individual variances and sample sizes.
• The degrees of freedom for the unpooled t-test are calculated using the Welch-Satterthwaite equation.
• The unpooled t-test formula includes the calculation of pooled variance.

  False
• If variances are suspected to be unequal, the unpooled t-test is preferred.

  True
• Match the concept with its definition:

  Confidence Interval ↔️ Likely to contain the true population parameter

  Significance ↔️ Probability observed difference occurred by chance
• Smaller p-values indicate stronger evidence against the null hypothesis.


  True
• The interpretation of a confidence interval is that the true parameter is in the interval with a specified level of confidence.


  True
• If the p-value is less than the significance level, the null hypothesis is rejected.


  True
• The degrees of freedom for the unpooled t-test are calculated using the formula $df =$ \frac{\left(\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}\right)^{2}}{\frac{\left(\frac{s_{1}^{2}}{n_{1}}\right)^{2}}{n_{1} - 1} + \frac{\left(\frac{s_{2}^{2}}{n_{2}}\right)^{2}}{n_{2} - 1}}
• Steps for calculating a confidence interval using a pooled t-test

  1️⃣ Check assumptions (Normality, Independence, Equal Variances)

  2️⃣ Calculate sample statistics

  3️⃣ Calculate pooled variance

  4️⃣ Determine degrees of freedom

  5️⃣ Find t-value

  6️⃣ Calculate confidence interval

  7️⃣ Interpret the interval
• The formula for calculating the pooled variance in a pooled t-test is s_{p}^{2}