AP Statistics Chapter 10

Pooled two-sample t statistic

A special version of the two-sample t statistic that assumes that the two populations have the same variance.

Sampling distribution of x var 1 - x var 2

Choose an SRS of size 1 n from population 1 with mean μ1 and standard deviation σ1 and an independent SRS of size 2 n from population 2 with mean μ2 and standard deviation σ 2 . • Shape: When the population distributions are Normal, the sampling distribution of 1 2 x x − is Normal. In other cases, the sampling distribution of 1 2 x x − will be approximately Normal if the sample sizes are large enough ( 1 n ≥ 30 and 2 n ≥ 30 ). • Center: The mean of the sampling distribution is μ μ 1 2 − . That is, the difference in sample means is an unbiased estimator of the difference in population means. • Spread: The standard deviation of the sampling distribution of 1 2 x x − is 2 2 1 2 1 2 n n σ σ + as long as each sample is no more than 10% of its population (the 10% condition).

Sampling distribution of p hat 1 - p hat 2

Choose an SRS of size 1 n from population 1 with proportion of successes 1 p and an independent SRS of size 2 n from population 2 with proportion of successes 2 p . • Shape: When np n p np n p 11 1 1 2 2 2 2 , 1 , , and 1 ( − − ) ( ) are all at least 10, the sampling distribution of 1 2 p p ˆ ˆ − is approximately Normal. • Center: The mean of the sampling distribution is 1 2 p p − . That is, the difference in sample proportions is an unbiased estimator of the difference in population proportions. • Spread: The standard deviation of the sampling distribution of 1 2 p p ˆ ˆ − is 1 12 2 1 2 p pp p ˆ (1 ) (1 ) n n − − + as long as each sample is no more than 10% of its population (the 10% condition).

Two-sample z test for the difference between two proportions

Suppose the Random, Normal, and Independent conditions are met. To test the hypothesis 01 2 Hpp : 0 − = , first find the pooled proportion ˆCp of successes in both samples combined. Then compute the z statistic ( ) ( ) ( ) 1 2 1 2 ˆ ˆ 0 ˆ ˆˆ ˆ 1 1 C CC C p p z p pp p n n − − = − − + Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the alternative hypothesis Ha

Two-sample t test for the difference between two means

Suppose the Random, Normal, and Independent conditions are met. To test the hypothesis 01 2 H : hypothesized value μ μ − = , compute the two-sample t statistic ( 12 1 2 ) ( ) 2 2 1 2 1 2 s x x t s n n −−− μ μ = + Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis Ha . Use the t distribution with degrees of freedom approximated by technology or the smaller of 1 n −1 and 2 n −1.

Standard error of p hat 1 - p hat 2

The estimated standard deviation of the statistic 1 2 p p ˆ ˆ − , given by 12 2 1 2 ppp p ˆˆˆ ˆ (1 ) (1 ) n n

Standard error of x var 1 - x var 2

The estimated standard deviation of the statistic 1 2 x x − , given by 2 2 1 2 1 2 s s n n

Two-sample z interval for a difference between two proportions

When the Random, Normal, and Independent conditions are met, an approximate level C confidence interval for 1 2 p p ˆ ˆ − is ( ) 1 12 2 ( ) ( ) * 1 2 1 2 ˆ ˆˆ ˆ 1 1 ˆ ˆ p pp p pp z n n − − −± + where z* is the critical value for the standard Normal curve with area C between -z* and z*.

Two-sample t interval for a difference between two means

When the Random, Normal, and Independent conditions are met, an approximate level C confidence interval for 1 2 x x − is ( ) 2 2 * 1 2 1 2 1 2 s s xx t n n −± + Here t* is the critical value for confidence level C for the t distribution with degrees of freedom from either Option 1 (technology) or Option 2 (the smaller of 1 n −1 and 2 n −1).

Two-sample t statistic

When we standardize the estimate 1 2 x x − , the result is the two-sample t statistic ( 12 1 2 ) ( ) 2 2 1 2 1 2 x x t s s n n −−− μ μ = + The statistic t has the same interpretation as any z or t statistic: it says how far 1 2 x x − is from its mean in standard deviation units.

Randomization distribution

the distribution of a statistic in repeated random assignments o experimental units to treatment groups assuming that the specific treatment received doesn't affect individual responses. when the random, normal, and independent conditions are met, our usual inference procedures based on the sampling distribution of the statistic will be approximately correct

Pooled (combined) sample proportion

the overall proportion of successes in the two samples is the count of successes in both samples combined / count of individuals in both samples combined