Hypothesis Testing: 1-Tailed, Small Samples, Proportions

Hypothesis Testing: 1-Tailed, Small Samples, Proportions BUSA 2100 Sec. 9.1 (continued), 9.3, 9.5, 9.6

One-Tailed Hypothesis Tests • Two-tailed tests allow for differences in either direction, but some problems lend themselves to one-tailed tests. • Ex. 1: A manufacturer claims that its 2-liter bottles contain at least 67.6 oz. • Follow the legal analogy; assume claim is true; give the benefit of the doubt. • H0: mu >= 67.6; Ha: mu < 67.6 .

One-Tailed Tests, Page 2 • Draw the acceptance and rejection regions and explain. • If H0 is accepted, the claim won’t be challenged. • If H0 is rejected, the claim is believed to be incorrect, and action is taken (adjust the filling equipment).

One-Tailed Tests, Page 3 • Example 2: A particular type of car currently averages 25 mpg. • A product research group has designed a new fuel-injection system that it hopes will increase mpg. • The group is trying to determine if an increase occurs. • We want strong evidence that an increase occurs.

One-Tailed Tests, Page 4 • Let the null hypothesis be the opposite of the research hypothesis. • H0: mu <= 25; Ha: mu > 25. • Draw accept/reject regions. If H0 is accepted, evidence is not conclusive. • If H0 is rejected, the research hyp. is accepted; mpg has been significantly increased; produce the new system.

One-Tailed Problem (Hotel Rates) • Two possibilities for 1-tailed tests. We’ll always put = with the null hypothesis. • Example 3: A luxury hotel chain claims that the mean weekend bill for a family is $350 or less. Test this claim at the .01 level. • Step 1:

Room Rate Problem, Page 2 • The rejection area always relates to the alternative hypothesis. • Step 2: • In a sample of 14, X-bar = $360, s = $45 • Step 3:

Room Rate Problem, Page 3 • Steps 4 and 5:

Hypothesis Testing Example (Tires) • Example 4: A research group hopes a new type of radial tire will have a mean life of 50,000 miles or more. Do a hypothesis test at the .05 level. • Step 1:

Tire Example, Page 2 • Step 2: • Sample results: In a sample of 100, X-bar = 50,600, s = 3,000. • Step 3:

Tire Example, Page 3 • Steps 4 and 5:

Hypothesis Testing Example (Plant Food) • Example 5: A new plant food has been designed to increase height of plants. • The plant food is tested on a sample of 12 plants. Heights are normally distrib-uted and the usual mean is 18 inches. • Using alpha = .05, is there sufficient reason to believe that the plant food increases height?

Plant Food Example, Page 2 • Steps 1 and 2:

Plant Food Example, Page 3 • Sample results, X-bar=19.5, s= 3. • Steps 3 - 5:

Plant Food Example, Page 4 • Note the z-value for alpha = .05, 1-tailed is 1.65. So for n >= 100, we would have rejected H0 since 1.732 > 1.65 . • t requires stronger evidence because of a smaller, less accurate sample. • With small samples, we want to be even more sure that a hypothesis is wrong before rejecting it.

Hypothesis Testing for Proportions (Restaurant) • Example 6: In the past, 40% of Burger World’s customers used the drive-thru. • The drive-through has been redesigned. Has the proportion who use it changed? Test at the .10 level. • Step 1:

Restaurant Example, Page 2 • Step 2: • Sample results: Of 140 people, 63 used the drive-through. • Step 3:

Restaurant Example, Page 3 • Steps 4 and 5:

Professional Organization Hypothesis Testing Example • Example 7: In the past, a professional organization has had 20% females. • A promotion has been done to increase the proportion of females. • A sample of 400 contained 300 males and 100 females. • Has the proportion of females increased at the .01 level?

Professional Organization Example, Page 2 • Steps 1 – 3:

Professional Organization Example, Page 3 • Steps 4 and 5, and P-value.

Hypothesis Testing: 1-Tailed, Small Samples, Proportions