Chapter 10 Part 2

Sampling distribution of X1 - X2

choose and SRS of size n1 from population 1 with mean u1 and standard deviation O1 and an independent SRS of size n2 from population 2 with mean u2 and standard deviation O2. Shape: When the population distributions are Normal, the sampling distribution is normal. In other cases, the sampling distribution will be approximately normal if the sample sizes are large enough or greater than or equal to 30. Center: the mean of the sampling distribution is u1-u2 or the difference in sample means is an unbiased estimator of the difference in population means. Spread: the standard deviation of the sampling distribution is root(O1^2/n1 + O2^2/n2) as long as each sample is no more than 10% of its population

Sampling distribution of P1 - P2

choose and SRS of size n1 from population 1 with proportion of successes p1 and an independent SRS of size n2 from the population of successes p2. Shape: when np and nq are at least 10, the sampling distribution is approximately normal. Center: the mean of the sampling distribution is p1-p2 or the difference in the sample proportions is an unbiased estimator of the difference in population proportions. Spread: the standard deviation of the sampling distribution is root(p1q1/n1 + p2q2/n2) as long as each sample is no more than 10% of its population

Two-sample z test for the difference between two proportions

suppose the random, normal and independent conditions are met. to test the hypothesis, first find the pooled proportion Pc of successes in both samples combined. Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the Ha.

Two-sample t test for the difference between two means

suppose the random, normal, and independent conditions are met. to test the hypothesis Ho: u1-u2 = hypothesized value, compute the two-sample t statistic (formula sheet) find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the Ha. use the t distribution with degrees of freedom approximated by technology or the smaller of the degrees of freedom.

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

Standard error of X1-X2

the estimated standard deviation of the statistic given by root(s1^2/n1 + s2^2/n2)

Standard error of P1-P2

the estimated standard deviation of the statistic is given by root(p1q1/n1 + p2q2/n2)

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 p1-p2 is (formula sheet) 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 the difference in means is (Refer to formula sheet) where t* is the critical value for confidence level C for the distribution with degrees of freedom

Two-sample t statistic

when we standardize the estimate for the difference in means, the result is the two-sample t statistic (formula sheet) when the statistic t has the same interpretation as any z or t statistic: it says how far the difference is from its mean in standard deviation units