Confidence Intervals: Difference of Two Means - AP Statistics

No significant difference. Zero difference falls within plausible range of values.

The interval becomes wider. Greater variability increases uncertainty in the estimate.

Estimate the population means. Sample means provide point estimates of unknown population parameters.

Insensitive to violations of assumptions. Method performs well even when assumptions are moderately violated.

When variances are equal. Equal variance assumption justifies pooling sample variances.

$t$-distribution. Difference of means follows $t$-distribution under standard assumptions.

Larger variance increases standard error. More variability leads to greater uncertainty in estimates.

Significant difference. Zero is outside the interval, indicating meaningful difference.

$( \bar{x}_1 - \bar{x}_2 ) \pm t^* \times SE( \bar{x}_1 - \bar{x}_2 )$. General structure: point estimate plus/minus margin of error.

Defines the interval width based on confidence level. Determines margin of error based on desired confidence level.

Less precision in estimating the mean difference. Wide intervals provide less specific information about the difference.

Samples are drawn separately. No overlap or influence between the two sample groups.

95% chance the interval contains the true mean difference. Confidence refers to the method, not any specific interval.

Data are approximately normally distributed. Required for valid use of $t$-distribution methods.

Samples must be randomly selected. Ensures samples represent their respective populations without bias.

Higher confidence level widens the interval. Trade-off between confidence and precision in estimation.

Difference between two population means. Compares means from two independent populations or groups.

Leads to incorrect interval estimation. Using pooled method when variances differ gives incorrect results.

The standard error decreases. Larger samples provide more precise estimates with smaller error.

$t^*$ is the critical value from the $t$-distribution. Found from $t$-distribution table based on confidence level and degrees of freedom.

Standard error of the difference of the two means. Measures variability of the difference between sample means.

Pooled standard deviation. Combines both sample standard deviations when variances are equal.

$H_0: \bar{x}_1 - \bar{x}_2 = 0$. States no difference between the two population means.

Normality, independence, and random sampling. Essential assumptions for valid $t$-distribution inference.

$df = n_1 + n_2 - 2$. Total sample size minus 2 when using pooled variance.

Use the Welch-Satterthwaite equation. Complex formula accounts for unequal variances and sample sizes.

Use a $t$-table or calculator. Critical value approximately 2.228 for 95% confidence with df=10.

Large sample sizes. $t$-distribution approaches normal as sample sizes increase (CLT).

Equal variances. Allows combining sample variances for more efficient estimation.