Hypothesis Testing: 1-Tailed, Hypothesis Testing: 1-Tailed, Small Samples, ProportionsSmall Samples, Proportions

BUSA 2100

Sec. 9.1 (continued), 9.3, 9.5, 9.6

One-Tailed Hypothesis Tests 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 2One-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 3One-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 4One-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)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 2Room 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 3Room Rate Problem, Page 3 Steps 4 and 5:

Hypothesis Testing Example (Tires)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 2Tire Example, Page 2 Step 2:

Sample results: In a sample of 100, X-bar = 50,600, s = 3,000.

Step 3:

Tire Example, Page 3Tire Example, Page 3 Steps 4 and 5:

Hypothesis Testing Example Hypothesis Testing Example (Plant Food)(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 2Plant Food Example, Page 2

Steps 1 and 2:

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

Plant Food Example, Page 4Plant 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 Hypothesis Testing for Proportions (Restaurant)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 2Restaurant Example, Page 2 Step 2:

Sample results: Of 140 people, 63 used the drive-through.

Step 3:

Restaurant Example, Page 3Restaurant Example, Page 3 Steps 4 and 5:

Professional Organization Professional Organization Hypothesis Testing ExampleHypothesis 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 Professional Organization Example, Page 2Example, Page 2

Steps 1 – 3:

Professional Organization Professional Organization Example, Page 3Example, Page 3

Steps 4 and 5, and P-value.