Hypothesis Test II: t tests Test for one population mean Test for the difference of two population means

Hypothesis Tests for Population Parameters Population proportion Population mean Difference between two population means

5 Steps in Hypothesis Tests Step 1. Determine the null (Ho) and alternative (Ha) hypotheses Ho : the population parameter = a constant (called the null value) Ha : upper or lower or two-tailed

5 Steps in Hypothesis Tests Step 2. Find an appropriate test statistic and pre-set the level of significance (called a level) Test statistic = (Sample estimate – null value) (S.E. of sample estimate@ null value) a= 0.05 (or 0.01 or 0.10)

5 Steps in Hypothesis Tests Step 3. Check the conditions and find the p-value assuming Ho is true The calculation of the p-value depends on the Ha.

5 Steps in Hypothesis Tests Step 4. Decide whether or not reject Ho based on the p-value p-value > a  fail to reject Ho p-value < a  reject Ho (= accept Ha)

5 Steps in Hypothesis Tests Step 5. Report the result in the context of the situation Reject Ho  There is sufficient evidence to support (Ha). Fail to reject Ho  There is not sufficient evidence to support (Ha).

Hypothesis Test for a Population Mean Step 1. Ho: m = mo Three possible Ha’s: Ha: m = mo (two-tailed) or Ha: m < mo (lower-tailed) Ha: m > mo (upper-tailed)

Hypothesis Test for a Population Mean Step 2. test statistic

Hypothesis Test for a Population Mean Step 3: Condition: Must be a random sample and the data must be bell-shaped (or normal) When the above conditions are met, use t-Table to find p-value (the required df = n-1) This test is called “one-sample t-test.”

Examples The average cost of 30 randomly selected used cars in a used car lot was $5,000 with sample standard deviation of $1,000. Find the p-value for testing Ho: m = $4,500 vs. Ha: m = $4,500. Answer: t = 2.74 and 0.006 < p-value < 0.016

Examples Find the p-value for testing Ho: m = $4,500 vs. Ha: m > $4,500. Ho: m = $4,500 vs. Ha: m < $4,500.

Hypothesis Test for the Difference of Two Population Means Step 1. Ho: m1 = m2 Three possible Ha’s: Ha: m1 = m2 (two-tailed) or Ha: m1 < m2 (lower-tailed) Ha: m1 > m2 (upper-tailed)

Hypothesis Test for the Difference between Two Population Means Step 2. test statistic

Hypothesis Test for the Difference between Two Population Means Step 3: Must be two independent random samples; the both samples must be bell-shaped (or normal) When the above conditions are met, use t-Table to find the p-value (the required df is the smaller one of (n1-1) and (n2-1)) This test is called “Two-sample t-test.”

Examples The average waiting time for 50 students to connect to the college server from the dormitories was 5 seconds, while the average time to connect for 20 students living off-campus apartments was 10 seconds. Assume the times to connect in the dormitories and the apartments are both bell-shaped. The sample standard deviations were 2 seconds for dormitories and 8 seconds for apartments

Examples Find the p-value for testing Ho: m1 = m2 vs. Ha: m1 = m2 Answer: t = -2.76 1: dormitories; 2: apartments