8-4
SignificanceofofExperimental
ExperimentalResults
Results
8-4 Significance
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
8-4 Significance of Experimental Results
Warm Up
The box-and-whisker plot shows the test
scores in Mrs. Howard’s first period math
class.
1. Find the minimum, maximum, median,
and quartile values for the data.
minimum: 82; 1st quartile: 88; median: 90;
3rd quartile: 93; maximum: 98
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Warm Up : Continued
The following is a list of test scores from
Mrs. Howard’s second period math class: 82,
83, 85, 87, 87, 87, 89, 90, 91, 95, 97, 97.
2. Find the mean, rounded to the nearest whole
number.
89
3. Draw a box plot for the data.
70
Holt McDougal Algebra 2
80
90
100
8-4 Significance of Experimental Results
Objectives
Use simulations and hypothesis testing
to compare treatments from a
randomized experiment.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Vocabulary
hypothesis testing
null hypothesis
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Suppose you flipped a coin 20 times. Even if the
coin were fair, you would not necessarily get
exactly 10 heads and 10 tails. But what if you got
15 heads and 5 tails, or 20 heads and no tails? You
might start to think that the coin was not a fair
coin, after all.
Hypothesis testing is used to determine whether
the difference in two groups is likely to be caused
by chance.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
For example, when tossing a coin 20 times, 11
heads and 9 tails is likely to occur if the coin is fair,
but if you tossed 19 heads and 1 tail, you could say
it was not likely to be a fair coin. To understand
why, calculate the number of possible ways each
result could happen. There are 220 possible
sequences of flips. Of these, how many fit the
description ‘19 heads, 1 tails’ and how many fit the
description, ‘11 heads, 9 tails’?
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Since there are
167,960
20
= 8398 times as many
sequences that fit the latter description as the
first, the result ‘11 heads, 9 tails’ is 8398 times as
likely as the result of ‘19 heads, 1 tails’! Therefore,
it is very unlikely that a coin that flipped 19 heads
and only 1 tails was a fair coin.
However, that outcome, while unlikely, is still
possible. Hypothesis testing cannot prove that a
coin is unfair – it is still possible for a coin to come
up with 19 heads by chance, it is just very unlikely.
Therefore, you can only say how likely or unlikely a
coin is to be biased.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Hypothesis testing begins with an assumption
called the null hypothesis. The null hypothesis
states that there is no difference between the two
groups being tested. The purpose of hypothesis
testing is to use experimental data to test the
viability of the null hypothesis.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Helpful Hint
The word null means “zero,” so the null hypothesis
is that the difference between the two groups is
zero.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Example 1 : Analyzing a Controlled Experiment
A researcher is testing whether a certain medication for
raising glucose levels is more effective at higher doses.
In a random trial, fasting glucose levels of 5 patients
being treated at a normal dose (Group A) and 5
patients being treated at a high dose (Group B) were
recorded. The glucose levels in mmol/L are shown
below.
A. State the null hypothesis for the experiment.
The glucose levels of the drug will be the same for the
control group (A) and the treatment group (B).
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Example 1: Continued
B. Compare the results for the control group and the
treatment group. Do you think that the researcher
has enough evidence to reject the null hypothesis?
The minimum, maximum, median, and quartile
values are as shown in the diagram below. There is a
small difference in the two groups that is likely to be
caused by chance. If anything, the treatment group
actually shows a tendency toward higher glucose
levels. The researcher cannot reject the null
hypothesis, which means that the medication is
probably just as effective at the normal dose as it is
at the high dose.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Example 1: Continued
4.0
Holt McDougal Algebra 2
5.0
6.0
8-4 Significance of Experimental Results
Check It Out! Example 1
A teacher wants to know if students in her morning
class do better on a test than students in her
afternoon class. She compares the test scores of 10
randomly chosen students in each class.
Morning class: 76,81,71, 80,88,66,79,67,85,68
Afternoon class: 80,91,74,92,80,80,88,67,75,78
a. State the null hypothesis.
The students in the morning class will have
the same test scores as the students in the
afternoon class
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Check It Out! Example 1 continued
b. Compare the results of the two groups. Does the
teacher have enough evidence to reject the null
hypothesis?
Yes; there is a large difference in the test scores
of the two classes. The teacher does have
enough evidence to reject the null hypothesis, so
she can conclude that students in her afternoon
class perform better on tests.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Example 2 : Using a Z-Test
The same test prep company claims that its private
tutoring can boost scores to an average of 2000. In a
random sample of 49 students who were privately
tutored, the average was 1910, with a standard
deviation of 150. Is there enough evidence to reject
the claim?
The z–value is –4.2, and | z | > 1.96. So, there is
enough evidence to reject the null hypothesis. You
can say with 95% confidence that the company’s
claim about private tutoring is false.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Check It Out! Example 2
A tax preparer claims an average refund of $3000. In
a random sample of 40 clients, the average refund was
$2600, and the standard deviation was $300. Is there
enough evidence to reject his claim?
Calculate the Z-value:
-400
≈ 47.43
2600-3000
300
√40
≈ -8.43
The z–value is –8.43, and | z | > 1.96. So, there is
enough evidence to reject the claim of the tax
preparer.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Lesson Quiz: Part I
1. A software company is testing whether a new
interface decreases the time it takes to complete a
certain task. In a random trial, Group A used the
existing interface and Group B used the new one.
The times in seconds are given for the members of
each group.
Group A: 12, 16, 12, 15, 17, 9, 13, 14, 16, 14
Group B: 8, 12, 10, 14, 9, 10, 13, 13, 10, 14
State the null hypothesis for the experiment.
The task will take the same amt. of time for both
groups.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Lesson Quiz: Part II
Compare the results for Group A and Group B. Do you
think that there is enough evidence to reject the null
hypothesis?
0
10
20
The median of Group B is below the first
quartile of Group A. The company can probably
reject the null hypothesis.
Holt McDougal Algebra 2
8-4 Significance of Experimental Results
Lesson Quiz: Part III
2. To disprove a previous study that claims that
college graduates make an average salary of
$46,000, a researcher records the salaries of 50
graduates and finds that the sample mean is
$43,000, with a standard deviation of $4,500. What
is the z-value, and can she reject the null hypothesis?
–4.71; yes
Holt McDougal Algebra 2