Sampling for Differences in Sample Means - AP Statistics
Card 1 of 30
Identify the assumption regarding the samples for the sampling distribution of differences in means.
Identify the assumption regarding the samples for the sampling distribution of differences in means.
Tap to reveal answer
Samples should be independent. No relationship between the two samples.
Samples should be independent. No relationship between the two samples.
← Didn't Know|Knew It →
What condition should be met if the populations are not normally distributed?
What condition should be met if the populations are not normally distributed?
Tap to reveal answer
Sample sizes should be large. Central Limit Theorem compensates for non-normality.
Sample sizes should be large. Central Limit Theorem compensates for non-normality.
← Didn't Know|Knew It →
What does the Central Limit Theorem ensure for the sampling distribution of the difference in means?
What does the Central Limit Theorem ensure for the sampling distribution of the difference in means?
Tap to reveal answer
It will be approximately normal if sample sizes are large. Large samples ensure normal distribution shape.
It will be approximately normal if sample sizes are large. Large samples ensure normal distribution shape.
← Didn't Know|Knew It →
What is the formula for pooled variance when variances are assumed equal?
What is the formula for pooled variance when variances are assumed equal?
Tap to reveal answer
$S_p^2 = \frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1+n_2-2}$. 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.
← Didn't Know|Knew It →
What is the degree of freedom used for testing the difference in means with equal variances?
What is the degree of freedom used for testing the difference in means with equal variances?
Tap to reveal answer
$df = n_1 + n_2 - 2$. Total observations minus parameters estimated.
$df = n_1 + n_2 - 2$. Total observations minus parameters estimated.
← Didn't Know|Knew It →
What is the condition for using a two-sample t-test?
What is the condition for using a two-sample t-test?
Tap to reveal answer
Samples are independent and normally distributed or large. Independence and normality or large sample requirements.
Samples are independent and normally distributed or large. Independence and normality or large sample requirements.
← Didn't Know|Knew It →
What assumption is necessary for using pooled variance?
What assumption is necessary for using pooled variance?
Tap to reveal answer
The population variances are equal. Required for pooled variance calculation validity.
The population variances are equal. Required for pooled variance calculation validity.
← Didn't Know|Knew It →
What is the impact of increasing sample size on the standard error of the difference in means?
What is the impact of increasing sample size on the standard error of the difference in means?
Tap to reveal answer
Standard error decreases. Larger samples reduce sampling variability.
Standard error decreases. Larger samples reduce sampling variability.
← Didn't Know|Knew It →
Identify the test used for comparing the means when population variances are known.
Identify the test used for comparing the means when population variances are known.
Tap to reveal answer
Z-test. Known population parameters allow normal distribution use.
Z-test. Known population parameters allow normal distribution use.
← Didn't Know|Knew It →
Identify the test used for comparing the means when population variances are unknown.
Identify the test used for comparing the means when population variances are unknown.
Tap to reveal answer
T-test. Unknown parameters require t-distribution.
T-test. Unknown parameters require t-distribution.
← Didn't Know|Knew It →
What does a significant p-value indicate in the context of a hypothesis test for the difference in means?
What does a significant p-value indicate in the context of a hypothesis test for the difference in means?
Tap to reveal answer
Reject the null hypothesis. Evidence against null hypothesis of no difference.
Reject the null hypothesis. Evidence against null hypothesis of no difference.
← Didn't Know|Knew It →
Identify the condition under which you would use Welch's t-test.
Identify the condition under which you would use Welch's t-test.
Tap to reveal answer
When variances are not equal. Accounts for unequal population variances.
When variances are not equal. Accounts for unequal population variances.
← Didn't Know|Knew It →
What is the primary purpose of a two-sample t-test?
What is the primary purpose of a two-sample t-test?
Tap to reveal answer
To compare the means of two independent samples. Tests whether two groups have different means.
To compare the means of two independent samples. Tests whether two groups have different means.
← Didn't Know|Knew It →
What effect does increasing the sample size have on the variability of the sampling distribution?
What effect does increasing the sample size have on the variability of the sampling distribution?
Tap to reveal answer
Variability decreases. Standard error decreases with larger samples.
Variability decreases. Standard error decreases with larger samples.
← Didn't Know|Knew It →
Define the symbol $\bar{x}_1 - \bar{x}_2$ in the context of sampling distributions.
Define the symbol $\bar{x}_1 - \bar{x}_2$ in the context of sampling distributions.
Tap to reveal answer
Difference in sample means. Point estimate for comparing two population means.
Difference in sample means. Point estimate for comparing two population means.
← Didn't Know|Knew It →
Identify the term for the difference between sample means and population means.
Identify the term for the difference between sample means and population means.
Tap to reveal answer
Sampling error. Random variation between sample and population.
Sampling error. Random variation between sample and population.
← Didn't Know|Knew It →
What is a critical assumption when interpreting a two-sample t-test?
What is a critical assumption when interpreting a two-sample t-test?
Tap to reveal answer
The samples are randomly selected. Ensures valid probability calculations and inference.
The samples are randomly selected. Ensures valid probability calculations and inference.
← Didn't Know|Knew It →
What does a high p-value suggest in the context of a hypothesis test for difference in means?
What does a high p-value suggest in the context of a hypothesis test for difference in means?
Tap to reveal answer
Fail to reject the null hypothesis. Insufficient evidence to conclude difference exists.
Fail to reject the null hypothesis. Insufficient evidence to conclude difference exists.
← Didn't Know|Knew It →
What is the implication of a confidence interval that includes zero in a difference of means test?
What is the implication of a confidence interval that includes zero in a difference of means test?
Tap to reveal answer
There may be no significant difference. Zero difference is plausible value.
There may be no significant difference. Zero difference is plausible value.
← Didn't Know|Knew It →
Which test statistic is used when population standard deviations are known?
Which test statistic is used when population standard deviations are known?
Tap to reveal answer
Z-score. Population parameters known, normal distribution applies.
Z-score. Population parameters known, normal distribution applies.
← Didn't Know|Knew It →
Which test statistic is used when population standard deviations are unknown?
Which test statistic is used when population standard deviations are unknown?
Tap to reveal answer
T-score. Population parameters unknown, t-distribution needed.
T-score. Population parameters unknown, t-distribution needed.
← Didn't Know|Knew It →
Identify the hypothesis test used when sample sizes are small and variances are unknown.
Identify the hypothesis test used when sample sizes are small and variances are unknown.
Tap to reveal answer
T-test with adjusted degrees of freedom. Small samples need t-distribution for unknown variances.
T-test with adjusted degrees of freedom. Small samples need t-distribution for unknown variances.
← Didn't Know|Knew It →
What is the effect of a larger sample size on the power of a hypothesis test?
What is the effect of a larger sample size on the power of a hypothesis test?
Tap to reveal answer
Increases the power. Better chance of detecting true differences.
Increases the power. Better chance of detecting true differences.
← Didn't Know|Knew It →
Define the parameter $\text{SE}$ in the context of sampling distributions.
Define the parameter $\text{SE}$ in the context of sampling distributions.
Tap to reveal answer
Standard Error. Standard deviation of sampling distribution.
Standard Error. Standard deviation of sampling distribution.
← Didn't Know|Knew It →
What is the formula for calculating the confidence interval for the difference in means?
What is the formula for calculating the confidence interval for the difference in means?
Tap to reveal answer
$CI = (\bar{x}_1 - \bar{x}_2) \pm t^* \times \text{SE}$. Margin of error added and subtracted from difference.
$CI = (\bar{x}_1 - \bar{x}_2) \pm t^* \times \text{SE}$. Margin of error added and subtracted from difference.
← Didn't Know|Knew It →
What is the implication of a confidence interval that includes zero in a difference of means test?
What is the implication of a confidence interval that includes zero in a difference of means test?
Tap to reveal answer
There may be no significant difference. Zero difference is plausible value.
There may be no significant difference. Zero difference is plausible value.
← Didn't Know|Knew It →
Identify the test used for comparing the means when population variances are unknown.
Identify the test used for comparing the means when population variances are unknown.
Tap to reveal answer
T-test. Unknown parameters require t-distribution.
T-test. Unknown parameters require t-distribution.
← Didn't Know|Knew It →
What does a significant p-value indicate in the context of a hypothesis test for the difference in means?
What does a significant p-value indicate in the context of a hypothesis test for the difference in means?
Tap to reveal answer
Reject the null hypothesis. Evidence against null hypothesis of no difference.
Reject the null hypothesis. Evidence against null hypothesis of no difference.
← Didn't Know|Knew It →
What is the formula for pooled variance when variances are assumed equal?
What is the formula for pooled variance when variances are assumed equal?
Tap to reveal answer
$S_p^2 = \frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1+n_2-2}$. 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.
← Didn't Know|Knew It →
What is the degree of freedom used for testing the difference in means with equal variances?
What is the degree of freedom used for testing the difference in means with equal variances?
Tap to reveal answer
$df = n_1 + n_2 - 2$. Total observations minus parameters estimated.
$df = n_1 + n_2 - 2$. Total observations minus parameters estimated.
← Didn't Know|Knew It →