Download Lecture 5 - West Virginia University

Lecture 11
Dan Piett
STAT 211-019
West Virginia University
Last Week
 Introduction to Hypothesis Testing
 Hypothesis Tests for µ
 Large Sample
 Small Sample
 Hypothesis Tests for p
Overview
 Hypothesis Tests on a difference in means
 Hypothesis Tests on a difference in proportions
 The 2-sided alternative
Section 11.1
Hypothesis Tests on the Difference in Means
Difference in Means
 Previously we created confidence intervals for the difference
in two population means.
 Male Scores vs Female Scores
 This is the same idea we had when we did confidence
intervals
 Our same rules apply for determining large and small sample
hypothesis tests
Large Sample Hyp. Test (n & m > 20)
H0: µx - µy = 0 (Does not have to be 0, but almost always is)
2.
HA: µx - µy < 0 (µy is bigger) or µx - µy > 0 (µx is bigger) or µx - µy ≠ 0
3.
Alpha is .05 if not specified
4.
Test Statistic = Z =
5.
P-value will come from the normal dist. Table

For > alternative: P(z>Z)

For < alternative: P(z<Z)

For ≠ alternative:2*P(z>|Z|)
6.
Our decision rule will be to reject H0 if p-value < alpha
7.
We have (do not have) enough evidence at the .05 level to conclude that the
mean of group x is ______ (<, >, ≠) the mean of group y
Requires a large sample size for both groups and equal population standard
deviations for both groups. Also requires independent random samples.
1.
Example
 A college statistics professor conjectures that students with good high school
math backgrounds (2+ courses) perform better in a college statistics course
than students with a poor high school math background (<2 courses). He
randomly selects 35 students with a good math background and 45 students
with a poor math background, and records exam scores from a college statistics
course. Test the hypothesis that the mean score of the good background
students will be higher than the mean score of the poor math background
students. Use alpha = .10. The summary data is as follows:
Group
Mean
Standard
Deviation
Sample
Size
2+
84.2
10.2
35
<2
73.1
14.3
45
Small Sample Hyp. Test (n or m < 20)
H0: µx - µy = 0 (Does not have to be 0, but almost always is)
2.
HA: µx - µy < 0 (µy is bigger) or µx - µy > 0 (µx is bigger) or µx - µy ≠ 0
3.
Alpha is .05 if not specified
4.
Test Statistic = T =
5.
P-value will come from the t-dist. Table with df = n+m-2

For > alternative: P(t>|T|)

For < alternative: P(t>|T|)

For ≠ alternative: 2*P(t>|T|)
6.
Our decision rule will be to reject H0 if p-value < alpha
7.
We have (do not have) enough evidence at the .05 level to conclude that the
mean of group x is ______ (<, >, ≠) the mean of group y
Requires both distributions are approximately normal with equal standard
deviations. Also requires independent random samples.
1.
Example
 A researcher wishes to assess a “new” teaching method for
“slow learners”. A random sample of 8 students use the new
method, and a random sample of 12 students use the
“standard” teaching method. After 6 months, an exam is
administered to each student. Does the data indicate that the
new teaching method is preferable? Use alpha = .05. The
summary statistics are as follows:
Group
Mean
Standard
Deviation
Sample
Size
New
77.125
4.853
8
Standard
72.333
6.344
12
Section 11.2
Hypothesis Tests for Two Independent Population Proportions
Difference in Pop. Proportions
 We are again interested in the difference in the proportions
of two populations
 Proportion of A’s on Exam 1 vs. Proportion of A’s on Exam 2
 Much like all the other tests covered, the same rules apply in
Hypothesis Testing that were involved in Confidence Intervals
 Also we will only be considering the case where the above is
true, therefore we will only be interested in tests using Z as
the test statistic.
Hypothesis Tests on the difference of
Proportions
H0: p1 – p2 = # (usually 0)
2.
HA: p < # or p > # or p ≠ #
3.
Alpha is .05 if not specified
4.
Test Statistic = Z =
5.
P-value will come from the normal dist. Table

For > alternative: P(z>Z)

For < alternative: P(z<Z)

For ≠ alternative:2*P(z>|Z|)
6.
Our decision rule will be to reject H0 if p-value < alpha
7.
We have (do not have) enough evidence at the .05 level to conclude that the
proportion of group x is ______ (<, >, ≠) the proportion of group y
Requires conditions on np’s. Also requires independent random samples
1.
Examples
 American Cancer Society wants to determine if the
proportion of smokers in the population of Americans has
decreased over the decade preceding 2002. In 1992, a
random sample of 150 Americans showed 58 who smoked.
In 2002, a random sample of 200 Americans included 64 who
smoked. Does the data indicate that the proportion of
smokers has decreased over the past decade? Use alpha =
.05.
Section 11.3
The 2-sided alternative
Notes on 2 Sided Alternatives
 Up until this point all of our examples have had alternative
hypotheses of the form < or >.
 What about ≠?
 What we will do for this is take our previous p-values times 2
 We take the value that makes sense
 If our statistic is less than our null hypothesis value, we use a <
probability
 If our statistic is more than our null hypothesis value, we use a
> probability
Example
 The quality control manager at a sugar processing packaging
plant must make sure that two-pound bags of sugar actually
contain two pounds of sugar. He randomly selects 50 bags of
sugar and weighs their contents. The sample mean is 1.962
pounds with a sample std. dev of 0.160 pounds. Does this
data indicate that the mean weight of all bags of sugar is
different from 2 pounds? Use alpha = .05.