Difference of Two Means (Setup) - AP Statistics
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Which test is used for comparing two population means?
Which test is used for comparing two population means?
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Two-sample t-test. Tests whether two population means are significantly different.
Two-sample t-test. Tests whether two population means are significantly different.
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What is the impact of unequal sample sizes on hypothesis testing?
What is the impact of unequal sample sizes on hypothesis testing?
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Affects standard error calculation. Unequal sizes require adjusted standard error calculations.
Affects standard error calculation. Unequal sizes require adjusted standard error calculations.
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What is an assumption regarding the sampling distribution in hypothesis testing?
What is an assumption regarding the sampling distribution in hypothesis testing?
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It is approximately normal. Central Limit Theorem ensures normality for hypothesis testing.
It is approximately normal. Central Limit Theorem ensures normality for hypothesis testing.
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Identify the primary use of a t-test comparing two samples.
Identify the primary use of a t-test comparing two samples.
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Assess mean differences. Determines if observed differences are statistically significant.
Assess mean differences. Determines if observed differences are statistically significant.
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What does a confidence interval represent in hypothesis testing?
What does a confidence interval represent in hypothesis testing?
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Range of plausible values for parameter. Contains likely values for the true population parameter.
Range of plausible values for parameter. Contains likely values for the true population parameter.
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What is the significance of a p-value less than 0.05?
What is the significance of a p-value less than 0.05?
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Reject the null hypothesis. Evidence against null hypothesis is statistically significant.
Reject the null hypothesis. Evidence against null hypothesis is statistically significant.
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What is the primary assumption for the Z-test with two samples?
What is the primary assumption for the Z-test with two samples?
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Large sample size or known population variance. Central Limit Theorem makes normal approximation valid.
Large sample size or known population variance. Central Limit Theorem makes normal approximation valid.
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What is the pooled standard deviation used for?
What is the pooled standard deviation used for?
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Comparing means with equal variances. Assumes both populations have the same variance.
Comparing means with equal variances. Assumes both populations have the same variance.
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Explain the impact of using a 99% confidence level instead of 95%.
Explain the impact of using a 99% confidence level instead of 95%.
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Wider confidence interval. Higher confidence level requires wider interval to capture parameter.
Wider confidence interval. Higher confidence level requires wider interval to capture parameter.
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What is the effect of increasing sample size on standard error?
What is the effect of increasing sample size on standard error?
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Standard error decreases. Larger samples provide more precise estimates of parameters.
Standard error decreases. Larger samples provide more precise estimates of parameters.
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In hypothesis testing, what does $H_a$ represent?
In hypothesis testing, what does $H_a$ represent?
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Alternative hypothesis. Statement of the effect or difference we're testing for.
Alternative hypothesis. Statement of the effect or difference we're testing for.
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In hypothesis testing, what does $H_0$ represent?
In hypothesis testing, what does $H_0$ represent?
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Null hypothesis. Statement of no effect or no difference to be tested.
Null hypothesis. Statement of no effect or no difference to be tested.
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What determines the critical value in a hypothesis test for two means?
What determines the critical value in a hypothesis test for two means?
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Significance level and degrees of freedom. These parameters define the rejection region boundaries.
Significance level and degrees of freedom. These parameters define the rejection region boundaries.
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What is the name of the distribution used for small sample sizes?
What is the name of the distribution used for small sample sizes?
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t-distribution. Used when population standard deviation is unknown.
t-distribution. Used when population standard deviation is unknown.
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What is the purpose of the F-test in the context of two means?
What is the purpose of the F-test in the context of two means?
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Tests equality of variances. Determines if equal variance assumption is reasonable.
Tests equality of variances. Determines if equal variance assumption is reasonable.
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What statistical test is used when variances are assumed unequal?
What statistical test is used when variances are assumed unequal?
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Welch's t-test. Modified t-test that doesn't assume equal population variances.
Welch's t-test. Modified t-test that doesn't assume equal population variances.
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How is the difference of two means denoted in hypothesis testing?
How is the difference of two means denoted in hypothesis testing?
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$( \bar{x}_1 - \bar{x}_2 )$. Standard notation for the difference between two sample means.
$( \bar{x}_1 - \bar{x}_2 )$. Standard notation for the difference between two sample means.
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What is the formula for the test statistic in a Z-test for two means?
What is the formula for the test statistic in a Z-test for two means?
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$Z = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE}$. Uses standard normal distribution for large sample hypothesis testing.
$Z = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE}$. Uses standard normal distribution for large sample hypothesis testing.
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Identify the appropriate test for large sample sizes comparing two means.
Identify the appropriate test for large sample sizes comparing two means.
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Z-test. Normal distribution applies when sample sizes are large.
Z-test. Normal distribution applies when sample sizes are large.
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What does $SE$ stand for in hypothesis testing?
What does $SE$ stand for in hypothesis testing?
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Standard Error. Measures variability of the sampling distribution.
Standard Error. Measures variability of the sampling distribution.
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Which option describes a Type II error in hypothesis testing?
Which option describes a Type II error in hypothesis testing?
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Failing to reject $H_0$ when it is false. False negative error, missing a real difference between means.
Failing to reject $H_0$ when it is false. False negative error, missing a real difference between means.
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Which option describes a Type I error in hypothesis testing?
Which option describes a Type I error in hypothesis testing?
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Rejecting $H_0$ when it is true. False positive error, concluding difference when none exists.
Rejecting $H_0$ when it is true. False positive error, concluding difference when none exists.
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What is the assumption about variances in a two-sample t-test?
What is the assumption about variances in a two-sample t-test?
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Variances are assumed equal or unequal. Determines whether to use pooled or separate variance formulas.
Variances are assumed equal or unequal. Determines whether to use pooled or separate variance formulas.
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What is the pooled variance formula for two samples?
What is the pooled variance formula for two samples?
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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 for equal variance assumption.
$s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}$. Weighted average of sample variances for equal variance assumption.
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Find the degrees of freedom for a two-sample t-test with $n_1=10$ and $n_2=12$.
Find the degrees of freedom for a two-sample t-test with $n_1=10$ and $n_2=12$.
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$df = 20$. Calculated as $n_1 + n_2 - 2 = 10 + 12 - 2 = 20$.
$df = 20$. Calculated as $n_1 + n_2 - 2 = 10 + 12 - 2 = 20$.
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What is the condition for using a two-sample t-test?
What is the condition for using a two-sample t-test?
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Samples are independent and normally distributed. Required assumptions to ensure valid test results.
Samples are independent and normally distributed. Required assumptions to ensure valid test results.
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What is the significance level typically used in hypothesis testing?
What is the significance level typically used in hypothesis testing?
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$\text{alpha} = 0.05$. Common threshold for statistical significance in hypothesis testing.
$\text{alpha} = 0.05$. Common threshold for statistical significance in hypothesis testing.
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What is the name of the distribution used for small sample sizes?
What is the name of the distribution used for small sample sizes?
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t-distribution. Used when population standard deviation is unknown.
t-distribution. Used when population standard deviation is unknown.
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In hypothesis testing, what does $H_0$ represent?
In hypothesis testing, what does $H_0$ represent?
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Null hypothesis. Statement of no effect or no difference to be tested.
Null hypothesis. Statement of no effect or no difference to be tested.
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Explain the impact of using a 99% confidence level instead of 95%.
Explain the impact of using a 99% confidence level instead of 95%.
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Wider confidence interval. Higher confidence level requires wider interval to capture parameter.
Wider confidence interval. Higher confidence level requires wider interval to capture parameter.
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