Download 10.2

Chapter 10
Section 2
Hypothesis Tests for a Population Mean
Assuming the Population Standard
Deviation is Known
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 1 of 55
Chapter 10 – Section 2
● Learning objectives
1

2

3

4

5

Understand the logic of hypothesis testing
Test hypotheses about a population mean with σ
known using the classical approach
Test hypotheses about a population mean with σ
known using P-values
Test hypotheses about a population mean with σ
known using confidence intervals
Understand the difference between statistical
significance and practical significance
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 2 of 55
Chapter 10 – Section 2
● Learning objectives
1

2

3

4

5

Understand the logic of hypothesis testing
Test hypotheses about a population mean with σ
known using the classical approach
Test hypotheses about a population mean with σ
known using P-values
Test hypotheses about a population mean with σ
known using confidence intervals
Understand the difference between statistical
significance and practical significance
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 3 of 55
Chapter 10 – Section 2
● We have the outline of a hypothesis test, just not
the detailed implementation
● How do we quantify “unlikely”?
● How do we calculate Type I and Type II errors?
● What is the exact procedure to get to a do not
reject / reject conclusion?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 4 of 55
Chapter 10 – Section 2
● There are three equivalent ways to perform a
hypothesis test
● They will reach the same conclusion
● The methods
 The classical approach
 The P-value approach
 The confidence interval approach
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 5 of 55
Chapter 10 – Section 2
● The classical approach
 If the sample value is too many standard deviations
away, then it must be too unlikely
● The P-value approach
 If the probability of the sample value being that far
away is small, then it must be too unlikely
● The confidence interval approach
 If we are not sufficiently confident that the parameter
is likely enough, then it must be too unlikely
● Don’t worry … we’ll be explaining more
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 6 of 55
Chapter 10 – Section 2
● The three methods all begin the same way
 We have a null hypothesis, that the actual mean is
equal to a value μ0
 We have an alternative hypothesis
● The three methods all set up a criterion
 A criterion that quantifies “unlikely”
 That the actual mean is unlikely to be equal to μ0
 A criterion that determines what would be a do not
reject and what would be a reject
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 7 of 55
Chapter 10 – Section 2
● The three methods all need information
 We run an experiment
 We collect the data
 We calculate the sample mean
● The three methods all make the same
assumptions to be able to make the statistical
calculations
 That the sample is a simple random sample
 That the sample mean has a normal distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 8 of 55
Chapter 10 – Section 2
● In this section we assume that the population
mean σ is known (as in section 9.1)
● We can apply our techniques if either
 The population has a normal distribution
 Our sample size n is large (n ≥ 30)
● In those cases, the distribution of the sample
mean x is normal with mean μ and standard
deviation σ / √ n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 9 of 55
Chapter 10 – Section 2
● The three methods all compare the observed
results with the previous criterion
 Classical – how many standard deviations
 P-value – the size of the probability
 Confidence interval – inside or outside the interval
● If the results are unlikely based on these
criterion, then we say that the result is
statistically significant
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 10 of 55
Chapter 10 – Section 2
● The three methods all conclude similarly
 We do not reject the null hypothesis, or
 We reject the null hypothesis
● We reject the null hypothesis when the result is
statistically significant
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 11 of 55
Chapter 10 – Section 2
● We now will cover how each of the
 Classical
 P-value, and
 Confidence interval
approaches will show us how to conclude
whether the result is statistically significant or not
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 12 of 55
Chapter 10 – Section 2
● Learning objectives
1

2

3

4

5

Understand the logic of hypothesis testing
Test hypotheses about a population mean with σ
known using the classical approach
Test hypotheses about a population mean with σ
known using P-values
Test hypotheses about a population mean with σ
known using confidence intervals
Understand the difference between statistical
significance and practical significance
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 13 of 55
Chapter 10 – Section 2
● The classical approach
● We compare the sample mean x to the
hypothesized population mean μ0
 Measure the difference in units of standard deviations
 A lot of standard deviations is far … few standard
deviations is not far
 Just like using a general normal distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 14 of 55
Chapter 10 – Section 2
● How far is too far?
● For example, we can use α = 0.05 as the level of
significance
● “Unlikely” means that this difference occurs with
probability α = 0.05 of the time, or less
● This concept applies to two-tailed tests, lefttailed tests, and right-tailed tests
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 15 of 55
Chapter 10 – Section 2
● For two-tailed tests
 The least likely 5% is the lowest 2.5% and highest
2.5% (below –1.96 and above +1.96 standard
deviations) … –1.96 and +1.96 are the critical values
 The region outside this is the rejection region
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 16 of 55
Chapter 10 – Section 2
● For left-tailed tests
 The least likely 5% is the lowest 5% (below –1.645
standard deviations) … –1.645 is the critical value
 The region less than this is the rejection region
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 17 of 55
Chapter 10 – Section 2
● For right-tailed tests
 The least likely 5% is the highest 5% (above 1.645
standard deviations) … +1.645 is the critical value
 The region greater than this is the rejection region
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 18 of 55
Chapter 10 – Section 2
● An example of a two-tailed test
● A bolt manufacturer claims that the diameter of
the bolts average 10.0 mm
 H0: Diameter = 10.0
 H1: Diameter ≠ 10.0
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is 0.3 mm
 The sample mean is 10.12 mm
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 19 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?





10.12 is 0.12 higher than 10.0
The standard error is (0.3 / √ 40) = 0.047
The test statistic is 2.53
The critical normal value, for α/2 = 0.025, is 1.96
2.53 is more than 1.96
● Our conclusion
 We reject the null hypothesis
 We have sufficient evidence that the population mean
diameter is not 10.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 20 of 55
Chapter 10 – Section 2
● An example of a left-tailed test
● A car manufacturer claims that the mpg of a
certain model car is at least 29.0
 H0: MPG = 29.0
 H1: MPG < 29.0
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is 0.5
 The sample mean mpg is 28.89
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 21 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?




28.89 is 0.11 lower than 29.0
The standard error is (0.5 / √ 40) = 0.079
The test statistic is -1.39
-1.39 is greater than -1.645, the left-tailed critical
value for α = 0.05
● Our conclusion
 We do not reject the null hypothesis
 We have insufficient evidence that the population
mean mpg is less than 29.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 22 of 55
Chapter 10 – Section 2
● An example of a right-tailed test
● A bolt manufacturer claims that the defective
rate of their product is at most 1.70 per 1,000
 H0: Defect Rate = 1.70
 H1: Defect Rate > 1.70
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is .06
 The sample defect rate is 1.78
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 23 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?




1.78 is 0.08 higher than 1.70
The standard error is (0.06 / √ 40) = 0.009
The test statistic is 8.43
8.43 is more than 1.645, the right-tailed critical value
for α = 0.05
● Our conclusion
 We reject the null hypothesis
 We have sufficient evidence that the population mean
rate is more than 1.70
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 24 of 55
Chapter 10 – Section 2
● Two-tailed test
 The critical values are zα/2 and –zα/2
 The rejection region is {less than –zα/2} and {greater
than z1-α/2}
● Left-tailed test
 The critical value is –zα
 The rejection region is {less than –zα}
● Right-tailed test
 The critical value is zα
 The rejection region is {greater than zα}
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 25 of 55
Chapter 10 – Section 2
● The difference is
x  0
● In units of standard deviations, this is
z0 
x  0
/ n
● This is called the test statistic
● If the test statistic is in the rejection region – we
reject
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 26 of 55
Chapter 10 – Section 2
● The general picture for a level of significance α
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 27 of 55
Chapter 10 – Section 2
● Learning objectives
1

2

3

4

5

Understand the logic of hypothesis testing
Test hypotheses about a population mean with σ
known using the classical approach
Test hypotheses about a population mean with σ
known using P-values
Test hypotheses about a population mean with σ
known using confidence intervals
Understand the difference between statistical
significance and practical significance
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 28 of 55
Chapter 10 – Section 2
● The P-value is the probability of observing a
sample mean that is as or more extreme than
the observed
● The probability is calculated assuming that the
null hypothesis is true
● We use the P-value to quantify how unlikely the
sample mean is
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 29 of 55
Chapter 10 – Section 2
● Just like in the classical approach, we calculate
the test statistic
x  0
z0 
/ n
● We then calculate the P-value, the probability
that the sample mean would be this, or more
extreme, if the null hypothesis was true
● The two-tailed, left-tailed, and right-tailed
calculations are slightly different
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 30 of 55
Chapter 10 – Section 2
● For the two-tailed test, the “unlikely” region are
values that are too high and too low
● Small P-values corresponds to situations where
it is unlikely to be this far away
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 31 of 55
Chapter 10 – Section 2
● For the left-tailed test, the “unlikely” region are
values that are too low
● Small P-values corresponds to situations where
it is unlikely to be this low
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 32 of 55
Chapter 10 – Section 2
● For the right-tailed test, the “unlikely” region are
values that are too high
● Small P-values corresponds to situations where
it is unlikely to be this high
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 33 of 55
Chapter 10 – Section 2
● For all three models (two-tailed, left-tailed, righttailed)
 The larger P-values mean that the difference is not
relatively large … that it’s not an unlikely event
 The smaller P-values mean that the difference is
relatively large … that it’s an unlikely event
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 34 of 55
Chapter 10 – Section 2
● Larger P-values
 A P-value of 0.30, for example, means that this value,
or more extreme, could happen 30% of the time
 30% of the time is not unusual
● Smaller P-values
 A P-value of 0.01, for example, means that this value,
or more extreme, could happen only 1% of the time
 1% of the time is unusual
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 35 of 55
Chapter 10 – Section 2
● The decision rule is
● For a significance level α
 Do not reject the null hypothesis if the P-value is
greater than α
 Reject the null hypothesis if the P-value is less than α
● For example, if α = 0.05
 A P-value of 0.30 is likely enough, compared to a
criterion of 0.05
 A P-value of 0.01 is unlikely, compared to a criterion
of 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 36 of 55
Chapter 10 – Section 2
● An example of a two-tailed test
● A bolt manufacturer claims that the diameter of
the bolts average 10.0 mm
 H0: Diameter = 10.0
 H1: Diameter ≠ 10.0
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is 0.3 mm
 The sample mean is 10.12 mm
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 37 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?




10.12 is 0.12 higher than 10.0
The standard error is (0.3 / √ 40) = 0.047
The test statistic is 2.53
The 2-sided P-value of 2.53 is 0.01 < 0.05 = α
● Our conclusion
 We reject the null hypothesis
 We have sufficient evidence that the population mean
diameter is not 10.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 38 of 55
Chapter 10 – Section 2
● An example of a left-tailed test
● A car manufacturer claims that the mpg of a
certain model car is at least 29.0
 H0: MPG = 29.0
 H1: MPG < 29.0
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is 0.5
 The sample mean mpg is 28.89
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 39 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?




28.89 is 0.11 lower than 29.0
The standard error is (0.5 / √ 40) = 0.079
The test statistic is -1.39
The 1-sided P-value of -1.39 is 0.08 > 0.05 = α
● Our conclusion
 We do not reject the null hypothesis
 We have insufficient evidence that the population
mean mpg is less than 29.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 40 of 55
Chapter 10 – Section 2
● An example of a right-tailed test
● A bolt manufacturer claims that the defective
rate of their product is at most 1.70 per 1,000
 H0: Defect Rate = 1.70
 H1: Defect Rate > 1.70
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is .06
 The sample defect rate is 1.78
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 41 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?




1.78 is 0.08 higher than 1.70
The standard error is (0.06 / √ 40) = 0.009
The test statistic is 8.43
The 1-sided P-value of 8.43 is extremely small
● Our conclusion
 We reject the null hypothesis
 We have sufficient evidence that the population mean
rate is more than 1.70
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 42 of 55
Chapter 10 – Section 2
● Compare the rejection regions for the classical
approach and the P-value approach
● They are the same
Classical
P-Value
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 43 of 55
Chapter 10 – Section 2
● Learning objectives
1

2

3

4

5

Understand the logic of hypothesis testing
Test hypotheses about a population mean with σ
known using the classical approach
Test hypotheses about a population mean with σ
known using P-values
Test hypotheses about a population mean with σ
known using confidence intervals
Understand the difference between statistical
significance and practical significance
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 44 of 55
Chapter 10 – Section 2
● The confidence interval approach yields the
same result as the classical approach and as the
P-value approach
● We compare
 A hypothesis test with a level of significance α
to
 A confidence interval with confidence (1 – α) •100%
● These are the same α’s
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 45 of 55
Chapter 10 – Section 2
● The relationship is
Not rejecting the
hypothesis
μ0 is inside the
Confidence interval
Rejecting the
hypothesis
μ0 is outside the
Confidence interval
● The hypothesis test calculation and the
confidence interval calculation are very similar
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 46 of 55
Chapter 10 – Section 2
● An example of a two-tailed test
● A bolt manufacturer claims that the diameter of
the bolts average 10.0 mm
 H0: Diameter = 10.0
 H1: Diameter ≠ 10.0
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
this measurement is 0.3 mm
 The sample mean is 10.12 mm
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 47 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?
 10.12 is 0.12 higher than 10.0
 The standard error is (0.3 / √ 40) = 0.047
 The confidence interval is 10.12 ± 1.96 • 0.047, or
10.03 to 10.21
 10.0 is outside (10.03, 10.21)
● Our conclusion
 We reject the null hypothesis
 We have sufficient evidence that the population mean
diameter is not 10.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 48 of 55
Chapter 10 – Section 2
● An example of a left-tailed test
● A car manufacturer claims that the mpg of a
certain model car is at least 29.0
 H0: MPG = 29.0
 H1: MPG < 29.0
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is 0.5
 The sample mean mpg is 28.89
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 49 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?
 28.89 is 0.11 lower than 29.0
 The standard error is (0.5 / √ 40) = 0.079
 The confidence interval limit is 28.89 + 1.645 • 0.079,
or 29.02
 29.0 is inside (–∞, 29.02)
● Our conclusion
 We do not reject the null hypothesis
 We have insufficient evidence that the population
mean mpg is less than 29.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 50 of 55
Chapter 10 – Section 2
● An example of a right-tailed test
● A bolt manufacturer claims that the defective
rate of their product is at most 1.70 per 1,000
 H0: Defect Rate = 1.70
 H1: Defect Rate > 1.70
● We take a sample of size 40
 (Somehow) We know that the standard deviation of
the population is .06
 The sample defect rate is 1.78
 We’ll use a level of significance α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 51 of 55
Chapter 10 – Section 2
● Do we reject the null hypothesis?
 1.78 is 0.08 higher than 1.70
 The standard error is (0.06 / √ 40) = 0.009
 The confidence interval limit is 1.78 – 1.645 • 0.009,
or 1.76
 1.70 is outside (1.76, ∞)
● Our conclusion
 We reject the null hypothesis
 We have sufficient evidence that the population mean
rate is more than 1.70
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 52 of 55
Chapter 10 – Section 2
● Learning objectives
1

2

3

4

5

Understand the logic of hypothesis testing
Test hypotheses about a population mean with σ
known using the classical approach
Test hypotheses about a population mean with σ
known using P-values
Test hypotheses about a population mean with σ
known using confidence intervals
Understand the difference between statistical
significance and practical significance
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 53 of 55
Chapter 10 – Section 2
● A significant statistical difference is one where
the hypothesis test, for equality, is rejected
● A statistical significance does not necessarily
mean that it is practically significant
● If we have a large sample size, we will be able to
pinpoint the rejectable values of the population
mean
● Our analysis may be unnecessarily precise
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 54 of 55
Summary: Chapter 10 – Section 2
● A hypothesis test of means compares whether
the true mean is either
 Equal to, or not equal to, μ0
 Equal to, or less than, μ0
 Equal to, or more than, μ0
● There are three equivalent methods of
performing the hypothesis test
 The classical approach
 The P-value approach
 The confidence interval approach
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 2 – Slide 55 of 55