Chapter 7 Hypothesis Testing

1
Chapter 7Hypothesis Testing

• 7-1 Basics of Hypothesis Testing
• 7-2 Testing a Claim about a Mean Large Samples
• 7-3 Testing a Claim about a Mean Small Samples
• 7-4 Testing a Claim about a Proportion
• 7- 5 Testing a Claim about a Standard
      Deviation (will cover with chap 8)

2
7-1

• Basics of
• Hypothesis Testing

3
Definition

• Hypothesis
• in statistics, is a statement regarding a
  characteristic of one or more populations

4
Steps in Hypothesis Testing

• Statement is made about the population
• Evidence in collected to test the statement
• Data is analyzed to assess the plausibility of
  the statement

5
Components of aFormal Hypothesis Test
6
Components of a Hypothesis Test

1. Form Hypothesis
2. Calculate Test Statistic
3. Choose Significance Level
4. Find Critical Value(s)
5. Conclusion

7
Null Hypothesis H0

• A hypothesis set up to be nullified or refuted
  in order to support an alternate hypothesis. When
  used, the null hypothesis is presumed true until
  statistical evidence in the form of a hypothesis
  test indicates otherwise.

8
Null Hypothesis H0

• Statement about value of population parameter
  like m, p or s
• Must contain condition of equality
• , ?, or ?
• Test the Null Hypothesis directly
• Reject H0 or fail to reject H0

9
Alternative Hypothesis H1

• Must be true if H0 is false
• ?, lt, gt
• opposite of Null
• sometimes used instead of

H1
Ha
10
Note about Forming Your Own Claims (Hypotheses)

• If you are conducting a study and want to use a
  hypothesis test to support your claim, the claim
  must be worded so that it becomes the
  alternative hypothesis.
• The null hypothesis must contain the condition
  of equality

11
Examples

• Set up the null and alternative hypothesis
• The packaging on a lightbulb states that the bulb
  will last 500 hours. A consumer advocate would
  like to know if the mean lifetime of a bulb is
  different than 500 hours.
• A drug to lower blood pressure advertises that it
  drops blood pressure by 20. A doctor that
  prescribes this medication believes that it is
  less. Set up the null and alternative
  hypothesis. (see hw 1)

12
Test Statistic

• a value computed from the sample data that is
  used in making the decision about the rejection
  of the null hypothesis
• Testing claims about the population proportion

x - µ
s
Z
n
13

• Critical Region - Set of all values of the test
  statistic that would cause a rejection of the
  null hypothesis
• Critical Value - Value or values that separate
  the critical region from the values of the test
  statistics that do not lead to a rejection of
  the null hypothesis

14
Critical Region and Critical Value

• One Tailed Test

Critical Region
Critical Value ( z score )
15
Critical Region and Critical Value

• One Tailed Test

Critical Region
Critical Value ( z score )
16
Critical Region and Critical Value

• Two Tailed Test

Critical Regions
Critical Value ( z score )
Critical Value ( z score )
17
Significance Level

• Denoted by ?
• The probability that the test statistic will
  fall in the critical region when the null
  hypothesis is actually true.
• Common choices are 0.05, 0.01, and 0.10

18
Two-tailed,Right-tailed,Left-tailed Tests

• The tails in a distribution are the extreme
  regions bounded
• by critical values.

19
Two-tailed Test

• H0 µ 100
• H1 µ ? 100

? is divided equally between the two tails of
the critical region
Means less than or greater than
Reject H0
Fail to reject H0
Reject H0
100
Values that differ significantly from 100
20
Right-tailed Test

• H0 µ ? 100
• H1 µ gt 100

Points Right
Values that differ significantly from 100
100
21
Left-tailed Test

• H0 µ ? 100
• H1 µ lt 100

Points Left
Fail to reject H0
Reject H0
Values that differ significantly from 100
100
22
Conclusions in Hypothesis Testing

• Traditional Method
• Reject H0 if the test statistic falls in the
  critical region
• Fail to reject H0 if the test statistic does not
  fall in the critical region
• P-Value Method
• Reject H0 if the P-value is less than or equal ?
• Fail to reject H0 if the P-value is greater than
  the ?

23
P-Value Methodof Testing Hypotheses

• Finds the probability (P-value) of getting a
  result and rejects the null hypothesis if that
  probability is very low
• Uses test statistic to find the probability.
• Method used by most computer programs and
  calculators.
• Will prefer that you use the traditional method
  on HW and Tests

24
Finding P-values

• Two tailed test
• p(zgta) p(zlt-a)
• One tailed test (right)
• p(zgta)
• One tailed test (left)
• p(zlt-a)

Where a is the value of the calculated test
statistic
Used for HW 3 5 see example on next two
slides
25
Determine P-value
Sample data x 105 or z 2.66
Reject H0 µ 100
Fail to Reject H0 µ 100


µ 73.4 or z 0
z 1.96
z 2.66
Just find p(z gt 2.66)
26
Determine P-value
Sample data x 105 or z 2.66
Reject H0 µ 100
Reject H0 µ 100
Fail to Reject H0 µ 100


z - 1.96
µ 73.4 or z 0
z 1.96
z 2.66
Just find p(z gt 2.66) p(z lt -2.66)
27
Conclusions in Hypothesis Testing

• Always test the null hypothesis
• Choose one of two possible conclusions
• 1. Reject the H0
• 2. Fail to reject the H0

28
Accept versus Fail to Reject

• Never accept the null hypothesis, we will fail
  to reject it.
• Will discuss this in more detail in a moment
• We are not proving the null hypothesis
• Sample evidence is not strong enough to warrant
  rejection (such as not enough evidence to convict
  a suspect guilty vs. not guilty)

29
Accept versus Fail to Reject
30
Conclusions in Hypothesis Testing

• Need to formulate correct wording of final
  conclusion

31
Conclusions in Hypothesis Testing

• Wording of final conclusion
• 1. Reject the H0
• Conclusion There is sufficient evidence to
  conclude (what ever H1 says)
• 2. Fail to reject the H0
• Conclusion There is not sufficient evidence to
  conclude (what ever H1 says)

32
Example

• State a conclusion
• The proportion of college graduates how smoke is
  less than 27. Reject Ho
• The mean weights of men at FLC is different from
  180 lbs. Fail to Reject Ho

Used for 6 on HW
33
Type I Error

• The mistake of rejecting the null hypothesis when
  it is true.
• ???(alpha) is used to represent the probability
  of a type I error
• Example Rejecting a claim that the mean body
  temperature is 98.6 degrees when the mean really
  does equal 98.6 (test question)

34
Type II Error

• the mistake of failing to reject the null
  hypothesis when it is false.
• ß (beta) is used to represent the probability of
  a type II error
• Example Failing to reject the claim that the
  mean body temperature is 98.6 degrees when the
  mean is really different from 98.6 (test
  question)

35
Type I and Type II Errors
True State of Nature
H0 False
H0 True
Reject H0
Correct decision
Type I error ?
Type II error ?
Decision
Fail to Reject H0
Correct decision
In this class we will focus on controlling a Type
I error. However, you will have one question on
the exam asking you to differentiate between the
two.
36
Type I and Type II Errors

• a p(rejecting a true null hypothesis)
• b p(failing to reject a false null hypothesis)
• n, a and b are all related

37
Example

• Identify the type I and type II error.
• The mean IQ of statistics teachers is greater
  than 120.
• Type I We reject the mean IQ of statistics
  teachers is 120 when it really is 120.
• Type II We fail to reject the mean IQ of
  statistics teachers is 120 when it really isnt
  120.

38
Controlling Type I and Type II Errors

• For any fixed sample size n , as ? decreases, ?
  increases and conversely.
• To decrease both ? and ?, increase the sample
  size.

39
Definition

• Power of a Hypothesis Test
• is the probability (1 - ??) of rejecting a false
  null hypothesis.
• Note No exam questions on this. Usually
  covered in a more advanced class in statistics.

40
7-2

• Testing a claim about the mean
• (large samples)

41
Traditional (or Classical) Method of Testing
Hypotheses

• Goal
• Identify a sample result that is significantly
  different from the claimed value
• By
• Comparing the test statistic to the critical value

42
Traditional (or Classical) Method of Testing
Hypotheses (MAKE SURE THIS IS IN YOUR NOTES)

• Determine H0 and H1. (and ? if necessary)
• Determine the correct test statistic and
  calculate.
• Determine the critical values, the critical
  region and sketch a graph.
• Determine Reject H0 or Fail to reject H0
• State your conclusion in simple non technical
  terms.

43
Test Statistic for Testing a Claim about a
Proportion
Can Use Traditional method Or P-value method
44
Three Methods Discussed

• 1) Traditional method
• 2) P-value method
• 3) Confidence intervals

45
Assumptions

• for testing claims about population means
• 1) The sample is a random sample.
• 2) The sample is large (n gt 30).
• a) Central limit theorem applies
• b) Can use normal distribution
• 3) If ? is unknown, we can use sample
  standard deviation s as estimate for ?.

46
Test Statistic for Claims about µ when n gt 30
x - µx
Z
?
n
47
Decision Criterion

• Reject the null hypothesis if the test statistic
  is in the critical region
• Fail to reject the null hypothesis if the test
  statistic is not in the critical region

48
Example A newspaper article noted that the mean
life span for 35 male symphony conductors was
73.4 years, in contrast to the mean of 69.5 years
for males in the general population. Test the
claim that there is a difference. Assume a
standard deviation of 8.7 years. Choose your own
significance level.
Step 1 Set up Claim, H0, H1

• Claim ? 69.5 years
• H0 ? 69.5
• H1 ? ? 69.5

Select if necessary ? level ?
0.05
49
Step 2 Identify the test statistic and
calculate
x - µ 73.4 69.5
z 2.65
?
8.7
n
35
50
Step 3 Determine critical region(s) and
critical value(s) Sketch
? 0.05
?/2 0.025 (two tailed test)
0.4750
0.4750
0.025
0.025
z - 1.96 1.96
Critical Values - Calculator
51
Step 4 Determine reject or fail to reject H0
Sample data x 73.4 or z 2.66
Reject H0 µ 69.5
Reject H0 µ 69.5
Fail to Reject H0 µ 69.5


z - 1.96
µ 73.4 or z 0
z 1.96
z 2.66
P-value P(z gt 2.66) x 2 .0078
REJECT H0
52
Step 5 Restate in simple nontechnical terms

• Claim ? 69.5 years
• H0 ? 69.5
• H1 ? ? 69.5
• There is sufficient evident to conclude that the
  mean life span of symphony conductors is
  different from the general population.
• OR
• There is sufficient evidence to conclude that
  mean life span of symphony conductors is
  different from 69.5 years.

REJECT
53
TI-83 Calculator

• Hypothesis Test using z (large sample)
• Press STAT
• Cursor to TESTS
• Choose ZTest
• Choose Input STATS
• Enter s and x and two tail, right tail or left
  tail
• Cursor to calculate or draw
• If your input is raw data, then input your raw
  data in L1 then use DATA

54
Testing Claims with Confidence Intervals

• We reject a claim that the population parameter
  has a value that is not included in the
  confidence interval
• Typically only used for two-tailed tests
• For one-tailed test the degree of confidence
  would be 1 2a (dont worry about this)

55
Testing Claims with Confidence Intervals
Claim mean age 69.5 years, where n 35, x
73.4 and s 8.7

• 95 confidence interval of 35 conductors (that
  is, 95 of samples would contain true
  value µ )
• 70.5 lt µ lt 76.3
• 69.5 is not in this interval
• Therefore it is very unlikely that µ 69.5
• Thus we reject claim µ 69.5 (same conclusion
  as previously stated)

56
7- 3

• Testing a claim about the mean
• (small samples)

57
Assumptions

• for testing claims about population means
  (student t distribution)
• 1) The sample is a random sample.
• 2) The sample is small (n ? 30).
• 3) The value of the population standard
  deviation ? is unknown.
• 4) population is approximately normal.

58
Test Statistic for a Student t-distribution
x -µx
t
s
n

• Critical Values
• Found in Table A-3
• Degrees of freedom (df) n -1
• Critical t values to the left of the mean are
  negative

59
Choosing between the Normal and Student
t-Distributions when Testing a Claim about a
Population Mean µ
Start
Use normal distribution with
x - µx
Is n gt 30 ?
Yes
Z??
?/ n
(If ? is unknown use s instead.)
No
Is the distribution of the population
essentially normal ? (Use a histogram.)
No
Use nonparametric methods, which dont require
a normal distribution.
Yes
Use normal distribution with
Is ? known ?
x - µx
Z??
?/ n
No
(This case is rare.)
Use the Student t distribution with
x - µx
t??
s/ n
60
Easier Decision Tree

• Use z if
• ? known or n is large
• Use t if
• is unknown and n is small and population is
  approximately normal
• MAKE SURE THIS IS IN YOUR NOTES

61
P-Value Method

• Table A-3 includes only selected values of ?
• Specific P-values usually cannot be found from
  table
• Use Table to identify limits that contain the
  P-value very confusing
• Some calculators and computer programs will find
  exact P-values

62
TI-83 Calculator

• Hypothesis Test using t (small sample)
• Press STAT
• Cursor to TESTS
• Choose TTest
• Choose Input STATS
• Enter s and x and two tail, right tail or left
  tail
• Cursor to calculate or draw
• If your input is raw data, then input your raw
  data in L1 then use DATA

63
Example

• Sample statistics of GPA include n20, x2.35 and
  s.7
• Test the claim that the GPA is greater than 2.0
• Use traditional method
• Use Calculator
• Find exact p-value (see excel TDIST function)

64
7-4

• Testing a claim about a proportion

65
Assumptions

• for testing claims about population proportions
• 1) The sample observations are a random sample.
• 2) The conditions for a binomial experiment are
  satisfied
• If np ? 5 and nq ? 5 are satisfied we 
• Use normal distribution to approximate binomial
  with µ np and ? npq

66
Notation
n number of trials
?
p x/n (sample proportion)

• p population proportion (used in the null
  hypothesis)
• q 1 - p

67
Test Statistic for Testing a Claim about a
Proportion
?
p - p
z
pq
n
68
?
p sometimes is given directly 10 of the
observed sports cars are red is expressed
as p 0.10
?
?
p sometimes must be calculated 96 surveyed
households have cable TV
and 54 do not is calculated using
?
x
96
p 0.64
n
(9654)

• (determining the sample proportion of households
  with cable TV)

69
CAUTION
?

• When the calculation of p results in a
  decimal with many places, store the number on
  your calculator and use all the decimals when
  evaluating the z test statistic.
• Large errors can result from rounding p too
  much.

?
70
Test Statistic for Testing a Claim about a
Proportion
?
Z
p - p
pq
n
x np
x - µ x - np n n p - p
?
z
?
pq
npq
npq
n
n
71
TI-83 Calculator

• Hypothesis Test using z (proportions)
• Press STAT
• Cursor to TESTS
• Choose 1-PropZTest
• Enter x and n and two tail, right tail or left
  tail
• Cursor to calculate or draw