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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: