Chapter 24

Comparing Means

Comparing Two Means

An educator believes that new reading activities for elementary school children will improve reading comprehension scores. She randomly assigns her third-grade students to one of two groups. The first group will use a traditional reading program and the second group will use the new reading activities. At the end of the experiment, both groups take a reading comprehension exam. Are the scores for the new reading activities group higher than for the traditional group?

Comparing Two Means

Look at boxplot of each group’s scores.

Comparing Two Means

What do you see? ____________________________________

____________________________________

____________________________________

____________________________________

Comparing Two Means

Does the new reading program produce better average scores?

For this particular class _________________________________

For population of all third-graders

_________________________________

Comparing Two Means

μ1 = _____________________________

μ2 = _____________________________

Interested in quantity μ1 - μ2.

Comparing Two Means

μ1 and μ2 are parameters (unknown).

________________________________

Estimate μ1 - μ2 with

Sampling Distribution

Assumptions: Random Samples Samples are Independent Nearly Normal Population Distributions

Sampling Distribution

If Assumptions hold, sampling distribution is

Sampling Distribution

σ1 and σ2 are parameters (unknown).

________________________________________________________

Sampling Distribution

Degrees of Freedom?

t really doesn’t have a t distribution.

The true distribution of t is ________________________________

when you use this formula for the degrees of freedom.

Degrees of Freedom?

2

2

22

2

2

1

21

1

2

2

22

1

21

11

11

ns

nns

n

ns

ns

df

Degrees of Freedom?

Problem: ____________________________________ ____________________________________

Can use simpler, more conservative method. ____________________________________

Inference for μ1 - μ2

C% confidence interval for μ1 - μ2

t* is critical value from t distribution table.d.f. = n1 – 1 or n2 – 1, whichever is smaller.

Example #1

A statistics student designed an experiment to test the battery life of two brands of batteries. For the sample of 6 generic batteries, the mean amount of time the batteries lasted was 206.0 minutes with a standard deviation of 10.3 minutes. For the sample of 6 name brand batteries, the mean amount of time the batteries lasted was 187.4 minutes with a standard deviation of 14.6 minutes. Calculate a 90% confidence interval for the difference in battery life between the generic and name brand batteries.

Example #1 (cont.)

Assumptions: Random samples

OK Independent samples

different batteries for each sample. Nearly Normal

data shows no real outliers.

Example #1 (cont.)

d.f. = 5μ1 = ______________________________

μ2 = ______________________________

6.14,4.187,6

3.10,0.206,6

222

111

syn

syn

Example #1 (cont.)

Example #1 (cont.)

Example #2

The Core Plus Mathematics Project was designed to help students improve their mathematical reasoning skills. At the end of 3 years, students in both the CPMP program and students in a traditional math program took an algebra test (without calculators). The 312 CPMP students had a mean score of 29.0 and a standard deviation of 18.8 while the 265 traditional students had a mean score of 38.4 with a standard deviation of 16.2. Calculate a 95% confidence interval for the mean difference in scores between the two groups.

Example #2 (cont.)

Assumptions: Random samples

no reason to think non-random Independent samples

different students in each group Nearly Normal

n1 and n2 are large, so not important.

Example #2 (cont.)

d.f. = smaller of 311 and 264 = 264μ1 = _____________________________

μ2 = _____________________________

2.16,4.38,265

8.18,0.29,312

222

111

syn

syn

Example #2 (cont.)

Example #2 (cont.)

Hypothesis Test for μ1 - μ2

HO: __________________________

HA: Three possibilities HA: ______________________________

HA: ______________________________

HA: ______________________________

Hypothesis Test for μ1 - μ2

Assumptions Random samples. Independent samples. Nearly Normal Population Distributions.

Hypothesis Test for μ1 - μ2

Test statistic:

d.f. = smaller of n1 – 1 and n2 – 1

P-value for Ha:__________________

P-value = P(t d.f. > t)

P-value for Ha: _________________

P-value = P(t d.f. < t)

P-value for Ha:__________________

P-value =

2*P(t d.f. > |t|)

Hypothesis Test for μ1 - μ2

Small p-value _____________________________________

_____________________________________

Large p-value _____________________________________ _____________________________________

Decision

If p-value < α

__________________________________

__________________________________

If p-value > α

__________________________________

__________________________________

Hypothesis Test for μ1 - μ2

Conclusion: Statement about value of μ1 - μ2. Always stated in terms of problem.

Example #1

Back to the reading example. The educator takes a random sample of all third graders in a large school district and divides them into the two groups. The mean score of the 18 students in the new activities group was 51.72 with a standard deviation of 11.71. The mean score of the 20 students in the traditional group was 41.8 with a standard deviation of 17.45. Is this evidence that the students in the new activities group have a higher mean reading score? Use α = 0.1.

Example #1 (cont.)

μ1 = ______________________________

μ2 = ______________________________

45.17,8.41,20

71.11,72.51,18

222

111

syn

syn

Example #1 (cont.)

HO: ____________

HA: ____________

Assumptions: Random Samples

OK Independent Samples

Different set of students in each group. Nearly Normal

boxplots look symmetric

Example #1 (cont.)

Example #1 (cont.)

d.f. = smaller of 17 or 19 = 17

P-value

Example #1 (cont.)

Decision:

Example #1 (cont.)

Conclusion:

Example #2

In June 2002, the Journal of Applied Psychology reported on a study that examined whether the content of TV shows influenced the ability of viewers to recall brand names of items featured in commercials. Researchers randomly assigned volunteers to watch either a program with violent content or a program with neutral content. Both programs featured the same 9 commercials. After the shows ended, subjects were asked to recall the brands in the commercials. Is there evidence that viewer memory for ads differs between programs? Use α = 0.05

Example #2 (cont.)

μ1 = _____________________________

μ2 = _____________________________

77.1,17.3,108

87.1,08.2,108

222

111

syn

syn

Example #2 (cont.)

HO: ______________

HA: ______________

Assumptions: Random Samples

no reason to think not random Independent Samples

Different people in each group. Nearly Normal

n1 and n2 are large so not important

Example #2 (cont.)

Example #2 (cont.)

d.f. = smaller of 107 or 107 = 107

P-value

Example #2 (cont.)

Decision:

Example #2 (cont.)

Conclusion: