Difference of Two Means (Test) - AP Statistics
Card 1 of 30
What is the consequence of violating the equal variance assumption?
What is the consequence of violating the equal variance assumption?
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May need to use a different test, like Welch's t-test. Adjusts for unequal variances between groups.
May need to use a different test, like Welch's t-test. Adjusts for unequal variances between groups.
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State the formula for the test statistic in a two-sample t-test.
State the formula for the test statistic in a two-sample t-test.
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$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\text{Difference})}{\text{SE}}$. Standardizes the observed difference by its standard error.
$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\text{Difference})}{\text{SE}}$. Standardizes the observed difference by its standard error.
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Identify the conditions for using a two-sample t-test.
Identify the conditions for using a two-sample t-test.
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Randomness, independence, and normality/large sample size. Ensures valid test assumptions are met.
Randomness, independence, and normality/large sample size. Ensures valid test assumptions are met.
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Which distribution is used for a two-sample t-test under the null hypothesis?
Which distribution is used for a two-sample t-test under the null hypothesis?
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Student's t-distribution. Accounts for uncertainty in variance estimates.
Student's t-distribution. Accounts for uncertainty in variance estimates.
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How do you calculate degrees of freedom for a two-sample t-test?
How do you calculate degrees of freedom for a two-sample t-test?
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Use the smaller of $n_1 - 1$ and $n_2 - 1$ or software for more precision. Conservative approach uses minimum; software gives exact calculation.
Use the smaller of $n_1 - 1$ and $n_2 - 1$ or software for more precision. Conservative approach uses minimum; software gives exact calculation.
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What is the pooled variance formula in a two-sample t-test?
What is the pooled variance formula in a two-sample t-test?
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$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$. Weighted average of sample variances.
$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$. Weighted average of sample variances.
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What does a significant p-value indicate in a two-sample t-test?
What does a significant p-value indicate in a two-sample t-test?
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Evidence against $H_0$, supporting $H_a$. Supports rejecting the null hypothesis.
Evidence against $H_0$, supporting $H_a$. Supports rejecting the null hypothesis.
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Define the p-value in the context of hypothesis testing.
Define the p-value in the context of hypothesis testing.
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Probability of observing data as extreme as the sample, assuming $H_0$ is true. Measures strength of evidence against null hypothesis.
Probability of observing data as extreme as the sample, assuming $H_0$ is true. Measures strength of evidence against null hypothesis.
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Identify one method to check for normality in a dataset.
Identify one method to check for normality in a dataset.
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Use a Q-Q plot or Shapiro-Wilk test. Assesses if data follows normal distribution.
Use a Q-Q plot or Shapiro-Wilk test. Assesses if data follows normal distribution.
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Choose an appropriate test for comparing means of more than two groups.
Choose an appropriate test for comparing means of more than two groups.
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ANOVA (Analysis of Variance). Extends two-sample comparison to multiple groups.
ANOVA (Analysis of Variance). Extends two-sample comparison to multiple groups.
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What does homoscedasticity mean in the context of t-tests?
What does homoscedasticity mean in the context of t-tests?
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Equal variances across groups being compared. Required assumption for valid t-test results.
Equal variances across groups being compared. Required assumption for valid t-test results.
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What is the role of the significance level ($\text{alpha}$) in hypothesis testing?
What is the role of the significance level ($\text{alpha}$) in hypothesis testing?
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Defines the threshold for rejecting the null hypothesis. Sets probability cutoff for statistical significance.
Defines the threshold for rejecting the null hypothesis. Sets probability cutoff for statistical significance.
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What assumptions are required for a two-sample z-test?
What assumptions are required for a two-sample z-test?
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Normality, known population variances, and random sampling. Uses z-distribution instead of t-distribution.
Normality, known population variances, and random sampling. Uses z-distribution instead of t-distribution.
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What is the impact of variance inequality on a two-sample t-test?
What is the impact of variance inequality on a two-sample t-test?
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May increase Type I error rate; consider using Welch's test. Welch's test adjusts for unequal variances.
May increase Type I error rate; consider using Welch's test. Welch's test adjusts for unequal variances.
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What is the formula for calculating the margin of error in a two-sample t-test?
What is the formula for calculating the margin of error in a two-sample t-test?
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$t^{*} \times SE$. Critical value multiplied by standard error.
$t^{*} \times SE$. Critical value multiplied by standard error.
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Define the power of a test in hypothesis testing.
Define the power of a test in hypothesis testing.
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Probability of correctly rejecting a false null hypothesis. Ability to detect true differences when they exist.
Probability of correctly rejecting a false null hypothesis. Ability to detect true differences when they exist.
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How does increasing sample size affect statistical power?
How does increasing sample size affect statistical power?
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Increases power, reducing the chance of a Type II error. More data improves ability to detect true effects.
Increases power, reducing the chance of a Type II error. More data improves ability to detect true effects.
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What is the formula for the standard error (SE) of the difference of means?
What is the formula for the standard error (SE) of the difference of means?
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$SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$. Combines variability from both samples.
$SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$. Combines variability from both samples.
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Which condition checks for independence in a two-sample t-test?
Which condition checks for independence in a two-sample t-test?
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Samples must be randomly selected from independent populations. Prevents bias from dependent observations.
Samples must be randomly selected from independent populations. Prevents bias from dependent observations.
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Identify one method to check for normality in a dataset.
Identify one method to check for normality in a dataset.
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Use a Q-Q plot or Shapiro-Wilk test. Assesses if data follows normal distribution.
Use a Q-Q plot or Shapiro-Wilk test. Assesses if data follows normal distribution.
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What is the role of the significance level ($\text{alpha}$) in hypothesis testing?
What is the role of the significance level ($\text{alpha}$) in hypothesis testing?
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Defines the threshold for rejecting the null hypothesis. Sets probability cutoff for statistical significance.
Defines the threshold for rejecting the null hypothesis. Sets probability cutoff for statistical significance.
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Which distribution is used for a two-sample t-test under the null hypothesis?
Which distribution is used for a two-sample t-test under the null hypothesis?
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Student's t-distribution. Accounts for uncertainty in variance estimates.
Student's t-distribution. Accounts for uncertainty in variance estimates.
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What is the purpose of a confidence interval in two-sample t-tests?
What is the purpose of a confidence interval in two-sample t-tests?
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To estimate the range for the true difference between population means. Provides range of plausible values for true difference.
To estimate the range for the true difference between population means. Provides range of plausible values for true difference.
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What does a 95% confidence interval imply?
What does a 95% confidence interval imply?
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95% of such intervals will contain the true population mean difference. Long-run frequency interpretation of confidence level.
95% of such intervals will contain the true population mean difference. Long-run frequency interpretation of confidence level.
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Identify a scenario where a paired t-test is more appropriate.
Identify a scenario where a paired t-test is more appropriate.
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When samples are dependent, such as pre-test/post-test designs. Accounts for correlation between paired observations.
When samples are dependent, such as pre-test/post-test designs. Accounts for correlation between paired observations.
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What does the central limit theorem imply for large samples?
What does the central limit theorem imply for large samples?
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Sample means are approximately normally distributed, regardless of the population distribution. Justifies normality assumption for hypothesis testing.
Sample means are approximately normally distributed, regardless of the population distribution. Justifies normality assumption for hypothesis testing.
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What is the relationship between sample size and the width of a confidence interval?
What is the relationship between sample size and the width of a confidence interval?
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Larger sample size leads to a narrower confidence interval. Increased precision with more data points.
Larger sample size leads to a narrower confidence interval. Increased precision with more data points.
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State the effect of an outlier on the two-sample t-test.
State the effect of an outlier on the two-sample t-test.
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Can significantly affect the test result, possibly leading to biased conclusions. Can distort mean and increase variability.
Can significantly affect the test result, possibly leading to biased conclusions. Can distort mean and increase variability.
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Choose an appropriate test for comparing means of more than two groups.
Choose an appropriate test for comparing means of more than two groups.
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ANOVA (Analysis of Variance). Extends two-sample comparison to multiple groups.
ANOVA (Analysis of Variance). Extends two-sample comparison to multiple groups.
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What is a Type II error in hypothesis testing?
What is a Type II error in hypothesis testing?
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Failing to reject a false null hypothesis. False negative error in hypothesis testing.
Failing to reject a false null hypothesis. False negative error in hypothesis testing.
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