Summary of Tests Confidence Limits Proportions and means

One sample Tests

Z test for a proportion Test statistic Null Hypothesis Alt. Hypothesis Critical Region H0: p = p0 H0: p ≠ p0 z < -za/2 or z > za/2 H0: p > p0 z > za H0: p < p0 z < -za

Confidence interval for a proportion (1 – a)100% confidence limits for p

Z test for the mean of a Normal Population Test statistic Null Hypothesis Alt. Hypothesis Critical Region H0: m = m 0 H0: m ≠ m 0 z < -za/2 or z > za/2 H0: m > m 0 z > za H0: m < m 0 z < -za

Confidence interval for a mean of a normal population (1 – a)100% confidence limits for m or for large samples

The t test for the mean of a Normal Population Test statistic Null Hypothesis Alt. Hypothesis Critical Region H0: m = m 0 H0: m ≠ m 0 t < -ta/2 or z > ta/2 H0: m > m 0 t > ta H0: m < m 0 t < -ta df = n -1

Confidence interval for a mean of a normal population (small samples) (1 – a)100% confidence limits for m df = n -1

Two sample Tests

Z test for a comparing two proportions Test statistic where Null Hypothesis Alt. Hypothesis Critical Region H0: p1 = p2 H0: p1 ≠ p2 z < -za/2 or z > za/2 H0: p1 > p2 z > za H0: p1 < p2 z < -za

Confidence intervals for the difference in two proportions (1 – a)100% confidence limits for p1 – p2

Z test for a comparing two means of Normal Populations Test statistic Null Hypothesis Alt. Hypothesis Critical Region H0: m1 = m2 H0: m1 ≠ m2 z < -za/2 or z > za/2 H0: m1 > m2 z > za H0: m1 < m2 z < -za

Confidence intervals for the difference in two means of normal populations (1 – a)100% confidence limits for m1 – m2 or for large samples

Test statistic where df = n + m - 2 The t test for a comparing two means of Normal Populations (variances assumed equal, sample sizes small) Test statistic where Null Hypothesis Alt. Hypothesis Critical Region H0: m1 = m2 H0: m1 ≠ m2 z < -za/2 or z > za/2 H0: m1 > m2 z > za H0: m1 < m2 z < -za df = n + m - 2

Confidence intervals for the difference in two means of normal populations (small sample sizes equal variances) (1 – a)100% confidence limits for m1 – m2 where

Tests, Confidence intervals for the difference in two means of normal populations (small sample sizes, unequal variances)

Consider the statistic For large sample sizes this statistic has standard normal distribution. For small sample sizes this statistic has been shown to have approximately a t distribution with

The approximate test for a comparing two means of Normal Populations (unequal variances) Test statistic Null Hypothesis Alt. Hypothesis Critical Region H0: m1 = m2 H0: m1 ≠ m2 t < -ta/2 or t > ta/2 H0: m1 > m2 t > ta H0: m1 < m2 t < -ta

Confidence intervals for the difference in two means of normal populations (small samples, unequal variances) (1 – a)100% confidence limits for m1 – m2 with

Review Assignment solutions

A third possible solution is to use the approximate t distribution with t0.025 = 2.080 and t0.005 = 2.831 for 21 df.

(1 – a)100% confidence limits for m1 – m2 are: 29.95 to 48.63 99% confidence limits 26.57 to 52.01