Download Hypothesis Testing

Hypothesis Testing
EQ: When is a result statistically significant?
Goal of HT
To assess the evidence provided by data about some claim
concerning a population
Recall
Recall from last chapter we need to set up our hypothesis
(including parameter and population definitions).
Example 1
In Illinois, a random sample of 85 eighth grade students
has a mean score of 285. The standard deviation for all
students who take the national mathematics assessment
test is 30. This test result prompts a state school
administrator to declare that the mean score for the state’s
eight graders on the examination is more than 275. Is
there evidence to support the administrator’s claim? (Use
95% confidence level).
µ is the mean score for eighth grade students on the national mathematics
assessment test
Null Hypothesis:
Alternative Hypothesis:
H o :   275
H A :   275
Performing Significance Tests
Set up Hypotheses
1.
a.
b.
Only use parameters
The null always has equality!
Calculate a test statistic.
3. Make a decision to reject/fail to reject based on p-values
4. Interpret the results
2.
Example 1: Ctd
1.
Set up Hypotheses
H o :   275
H A :   275
2.
Calculate a test statistic.
3.
Make a decision to reject/fail to reject based on p-values
Thus we reject the null hypothesis
4.
Interpret the results
There is evidence to support the administrator’s claim that the
mean test scores are greater than 275.
BUT WAIT!!!!
 Will this work for ALL cases??????
 What about if we don’t know the population standard
deviation???????
Example 2
Home Depot brand light bulbs state on the package “Average
Life 1000Hr.” A class believe this number to be high and
tested 60 randomly selected bulbs. For these 60 bulbs, the
average bulb life was 970 hours and the standard deviation
was 120 hours. Is there enough evidence to reject home
depots claim?
1. Set up Hypotheses:
Null Hypothesis:
H o :   1000
Alternative Hypothesis: H A :   1000
Test Statistic
Determined by the parameter and conditions
1 sample means
x
t
s
n
Conditions
1. SRS
2. Normal
****Use z if sigma is known
2.Test Statistic
t
x   970  1000

 1.936
s
120
n
60
Based on the data, we found a test statistic of -1.936.
How likely is it to get that value?
3. Make a decision to reject/fail to reject based on p-values
P-value
P (t  1.936)  .0288
There is a 2.88% chance of getting a test statistic less than or equal
to the one we got if the actual life of the bulbs is 1000 hours.
P-values
Remember: The SMALLER the p-value, the more evidence there is to reject the null
hypothesis
A rejection level can be set a specific level denoted by α,
if not stated use α=.05
From the example, the p-value was .0288
Decision: Reject the null since .0288 < .05
4. Interpret the results
It appears that Home Depot’s claim is false, the mean life
of the light bulbs is not 1000 hours
In an advertisement, a pizza shop claims that its mean delivery time is
less than 30 minutes. A random sample of 36 delivery times has a
sample mean of 28.5 minutes and a standard deviation of 3.5
minutes. Is there enough evidence to support the claim?
1. Hypotheses: H o :   30
H A :   30
2.Test Statistic: 1-sample mean t-test, 35 df
Conditions: SRS, Normal (n>30)
x   28.5  30
t

 2.571
s
3.5
n
36
3. Decision
P(t  2.571)  .0073
Reject the null
4. Interpret
The average delivery time appears to be less than 30 minute. The
company’s claim is correct.
2 Tail test: Example 4
A manager for a company reports an average of 150 sales per day.
His boss suspects that this averages is not accurate. He selects 35
days and determines the number of sales each day. The sample
mean is 143 daily sales with a standard deviation of 15 sales. At
the .01 level, is there evidence to doubt the mangers claim?
1. Hypotheses: H o :   150
H A :   150
When the alternative is not equal, it is a 2 tail test
2.Test Statistic: 1-sample mean t-test, 34 df
Conditions: SRS, Normal (n>30)
x
143  150
t

 2.761
s
15
n
35
3. Decision
We must consider both directions when finding the p-value
P(t  2.761 or t  2.761)  .0092
Since .0092<.01, reject the null
4. Interpret
The average number of sales does not appear to be 150. There is
evidence to support the boss’ claim.
Example 5
Tennis balls produced by a particular machine do not have identical
diameters because of variation in the manufacturing process. A
machine was initially calibrated to produce balls with a mean of 3
inches and a standard deviation of .25 inches. A random sample of 90
balls produced a mean of 3.05 inches. Is there evidence that the
diameter of the tennis balls is larger than it is supposed to be?
1. Hypotheses:
Ho :   3
HA :   3
2.Test Statistic: 1-sample mean z-test
Conditions: SRS, Normal (n>30)
z
x

n
3.05  3

 1.897
.25
90
3. Decision
P( z  1.897)  .0289
Since .0289>.01, fail to reject the null
4. Interpret
There is not enough evidence to find that the machine is making
tennis balls with diameters that are larger than specified.