Tests of Significance Chapter 11 Confidence intervals are used to estimate a population parameter. Tests of significance assess the evidence provided by.
Download ReportTranscript Tests of Significance Chapter 11 Confidence intervals are used to estimate a population parameter. Tests of significance assess the evidence provided by.
Slide 1
Tests of Significance
Chapter 11
Slide 2
Confidence intervals are used to estimate a population parameter.
Tests of significance assess the evidence provided by the data
about some claim concerning the population.
Example: I claim that I make 80% of my free throws. You do not
believe me, so we go to the gym and I shoot 20 free throws,
making only 8! You use this result as backing up your claim that
I do not make 80% of free throws, because if I truly made 80%
of free throws I would be very unlikely to make only 8. What is
the probability that I only make 8 if my claim was true?
The reasoning of significance tests, like that of
confidence intervals, is based on what we would expect
to happen if we repeated an experiment/sample many
times.
Slide 3
Example: Diet cola uses artificial sweeteners to avoid sugar. These
sweeteners gradually lose sweetness over time. To test a new diet drink
we have 10 tasters sample the drink, store the drink 4 months, and have
them taste the drink again. The drinks are scored on a sweetness scale
and the difference between the first and second taste is our score for the
loss of sweetness (matched pairs design). Large scores represent large
sweetness losses. Below are the scores:
2.0
0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3
Does the data provide good evidence that the
cola lost sweetness in storage? How can you
answer this question?
Slide 4
x 1.02
Is this large enough to be significant. A test of
significance asks:
Does the sample mean 1.02 reflect a real loss
of sweetness or could it just have occurred by
chance?
A test of significance starts with a careful
statement of these two alternatives. First
determine the parameter of interest. We
always draw conclusions about a population
parameter! In this case the population mean
μ is the mean loss of sweetness. Our 10
tasters provide a sample of this population.
Slide 5
Now we can state the null hypothesis. The null
hypothesis states that there is no effect or no change
in the population, i.e. drink did not lose sweetness:
H0 : 0 0
The effect that we suspect to be true will be our
alternative hypothesis. The alternative hypothesis
states that there is a change in the population, i.e.
drink did lose sweetness
H : 0 one-sided alternative
a
0
To answer the question we will use our knowledge of
how would behave in repeated samples. To be able
to do that we need to know the standard deviation of
the population. Let’s assume 1
Slide 6
Under the assumption that the null hypothesis
is true, we can now draw the following normal
curve:
-0.96
-0.64
-0.32
0
0.32
0.64
0.96
Label the curve based on the hypothesized mean, 0 and σ/sqrt(10)=0.32
Slide 7
Inference Toolbox
The medical director of a large company looks
at the medical records of 72 male executives
aged 35-44. He finds that their average blood
pressure is 126.07.
126.07 If the average blood
pressure of all males aged 35-44 is 128 with a
standard deviation of 15,
15 does this sample
show evidence that the blood pressure of
executives is different from the regular
population?
n=72 x 1 2 6 .0 7 μ = 128 σ = 15
Slide 8
Step 1: Hypotheses
• State the population of interest &
parameter being studied.
Population: all male executives between 35-44 years
of age in the company.
Parameter: μ = mean blood pressure
• State your hypotheses.
H0: μ=128
Ha: μ128
Slide 9
Step 2: Conditions/Procedure
• State the procedure you will be using.
One-sample z -test
• Verify the conditions required to use it.
SRS
Independence
Normality
• Once conditions are verified, state the
distribution
x ~ N ( x , x )
Slide 10
Step 3: Calculate test statistic &
p-value
• To calculate the p-value, you are
calculating, P(x >,<,or μ0)
z
x 0
n
Slide 11
Step 4: Interpretation
• Connect
Relate the p-value to the level.
• Conclude
Reject or fail to reject H0.
• Context
How does the conclusion affect the
claim?
Slide 12
SRS
• A simple random sample is the
cornerstone to making inferences.
• A srs is the necessary condition that
ensures that as much variability
between the individuals in the
population is “cancelled” out when the
individuals are randomly selected.
• This way, any conclusions can be
“inferred” to the population.
Slide 13
Normality
• Being able to satisfy the normality
condition enables you to:
• Use a known and well established
distribution.
• Calculate probabilities – a Chapter 2 topic
• To verify this condition:
• Sample Means, x
• Sample Proportions, p
Slide 14
Sample Means, x
You KNOW the population is
Normally Distributed.
x
You DON’T KNOW the shape of
the distribution of the
population
x
Tests of Significance
Chapter 11
Slide 2
Confidence intervals are used to estimate a population parameter.
Tests of significance assess the evidence provided by the data
about some claim concerning the population.
Example: I claim that I make 80% of my free throws. You do not
believe me, so we go to the gym and I shoot 20 free throws,
making only 8! You use this result as backing up your claim that
I do not make 80% of free throws, because if I truly made 80%
of free throws I would be very unlikely to make only 8. What is
the probability that I only make 8 if my claim was true?
The reasoning of significance tests, like that of
confidence intervals, is based on what we would expect
to happen if we repeated an experiment/sample many
times.
Slide 3
Example: Diet cola uses artificial sweeteners to avoid sugar. These
sweeteners gradually lose sweetness over time. To test a new diet drink
we have 10 tasters sample the drink, store the drink 4 months, and have
them taste the drink again. The drinks are scored on a sweetness scale
and the difference between the first and second taste is our score for the
loss of sweetness (matched pairs design). Large scores represent large
sweetness losses. Below are the scores:
2.0
0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3
Does the data provide good evidence that the
cola lost sweetness in storage? How can you
answer this question?
Slide 4
x 1.02
Is this large enough to be significant. A test of
significance asks:
Does the sample mean 1.02 reflect a real loss
of sweetness or could it just have occurred by
chance?
A test of significance starts with a careful
statement of these two alternatives. First
determine the parameter of interest. We
always draw conclusions about a population
parameter! In this case the population mean
μ is the mean loss of sweetness. Our 10
tasters provide a sample of this population.
Slide 5
Now we can state the null hypothesis. The null
hypothesis states that there is no effect or no change
in the population, i.e. drink did not lose sweetness:
H0 : 0 0
The effect that we suspect to be true will be our
alternative hypothesis. The alternative hypothesis
states that there is a change in the population, i.e.
drink did lose sweetness
H : 0 one-sided alternative
a
0
To answer the question we will use our knowledge of
how would behave in repeated samples. To be able
to do that we need to know the standard deviation of
the population. Let’s assume 1
Slide 6
Under the assumption that the null hypothesis
is true, we can now draw the following normal
curve:
-0.96
-0.64
-0.32
0
0.32
0.64
0.96
Label the curve based on the hypothesized mean, 0 and σ/sqrt(10)=0.32
Slide 7
Inference Toolbox
The medical director of a large company looks
at the medical records of 72 male executives
aged 35-44. He finds that their average blood
pressure is 126.07.
126.07 If the average blood
pressure of all males aged 35-44 is 128 with a
standard deviation of 15,
15 does this sample
show evidence that the blood pressure of
executives is different from the regular
population?
n=72 x 1 2 6 .0 7 μ = 128 σ = 15
Slide 8
Step 1: Hypotheses
• State the population of interest &
parameter being studied.
Population: all male executives between 35-44 years
of age in the company.
Parameter: μ = mean blood pressure
• State your hypotheses.
H0: μ=128
Ha: μ128
Slide 9
Step 2: Conditions/Procedure
• State the procedure you will be using.
One-sample z -test
• Verify the conditions required to use it.
SRS
Independence
Normality
• Once conditions are verified, state the
distribution
x ~ N ( x , x )
Slide 10
Step 3: Calculate test statistic &
p-value
• To calculate the p-value, you are
calculating, P(x >,<,or μ0)
z
x 0
n
Slide 11
Step 4: Interpretation
• Connect
Relate the p-value to the level.
• Conclude
Reject or fail to reject H0.
• Context
How does the conclusion affect the
claim?
Slide 12
SRS
• A simple random sample is the
cornerstone to making inferences.
• A srs is the necessary condition that
ensures that as much variability
between the individuals in the
population is “cancelled” out when the
individuals are randomly selected.
• This way, any conclusions can be
“inferred” to the population.
Slide 13
Normality
• Being able to satisfy the normality
condition enables you to:
• Use a known and well established
distribution.
• Calculate probabilities – a Chapter 2 topic
• To verify this condition:
• Sample Means, x
• Sample Proportions, p
Slide 14
Sample Means, x
You KNOW the population is
Normally Distributed.
x
You DON’T KNOW the shape of
the distribution of the
population
x