Download What is a sampling distribution of the mean?

Retain or Reject H0?
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Retain or Reject H0?
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Retain or Reject H0?
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Retain or Reject H0?
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Retain or Reject H0?
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Retain or Reject H0?
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Retain or Reject H0?
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Outcome
The Binomial Distribution is a sampling distribution
(for the sign test).
The sampling distribution of a statistic:
(1) gives all the values that the statistic can take and
(2) gives the probability of getting each value under the assumption that it
resulted from chance alone.
The sampling distribution is the chance distribution to which we compare
our obtained results.
N = 10 Events
If chance alone is responsible, what is the probability of getting 5
tails? 6 tails? 9 tails? 7 or more tails?
N = 10 Events
.0001
H1= This coin is weighted for tails.
H0= This coin is NOT weighted for tails.
Use an alpha of .05
.0001
N = 10 Events
.0001
H1= This coin is weighted.
H0= This coin is NOT weighted.
Use an alpha of .05
.0001
Notice: The Binomial Test/Sign Test is ONLY appropriate when
there are just two possibilities of outcome (for example
heads/tails, correct/incorrect, etc….
Binomial Test Questions:
1. Is a coin weighted if out of ten flips it ends up at heads 9 times?
2. Does a fertilizer work if 9 times out of ten it produces greater
growth than without it?
3. Are college graduates smarter than non-college graduates if out
of ten randomly sampled geniuses 9 of them happen to be college
graduates?
THERE WILL BE A COMMON SET OF STEPS
YOU WILL USE FOR ALL HYPOTHESIS TESTS:
Step 1: State your hypotheses.
Step 2: Calculate the critical value.
Step 3: Calculate the obtained statistic.
Step 4: Make a decision.
How about the following question:
It has been documented that in the general population, the average
number of hours slept a night by Americans is 8, with a standard
deviation of 4 hours. I collect sample data on a set of 16 college
students, and determine that their average number of hours
sleeping per night is 6 hours. Do college students on average sleep
less than the general population?
In other words, I am comparing a sample mean (my 16 students)
with my KNOWN population mean and asking the question, are
they different?
It has been documented that in the general population, the average
number of hours slept a night by Americans is 8, with a standard deviation
of 4 hours. I collect sample data on a set of 16 college students, and
determine that their average number of hours sleeping per night is 6 hours
(s=1). Do college students on average sleep less than the general
population?
In other words, I am comparing a sample mean (my 16 students) with a
population mean and asking the question, are they different?
m=8
s=4
population
sample
x=6
s=1
8
6
What is a sampling distribution
of the mean?
7
8
8
9
2
6
6
6
6
6
6
6
7
7
11
8
5
8
10
4
7
0
4
6
10
5
3
5
9
6
2
4
0
6
3
3
8
3
14
8
10
2
1
9
5
1
7
5
7
11
5
3
5
9
8
4
5
9
4
3
5
9
2
4
12
12
1
4
10
7
m=8
s=4
8
7
11
7
6
6
What is a sampling distribution
of the mean?
7
8
8
9
2
6
6
6
6
6
6
6
7
7
11
8
5
8
10
4
7
0
4
6
10
5
3
5
9
6
2
4
0
6
3
3
8
3
14
8
10
2
1
9
5
1
7
5
7
11
5
3
5
9
8
4
5
9
4
3
5
9
2
4
12
12
1
4
10
7
m=8
s=4
8
7
11
7
6
6…
It has been documented that in the general population, the average
number of hours slept a night by Americans is 8, with a standard
deviation of 4 hours. I collect sample data on a set of 16 college
students, and determine that their average number of hours
sleeping per night is 6 hours (s=1). Do college students on average
sleep less than the general population?
In other words, I am comparing a sample mean (my 16 students)
with a population mean and asking the question, are they different?
m=8
s=4
population
sample
x=6
s=1
8
6
What is a sampling distribution
of the mean?
_
x
7
8
8
9
2
6
6
6
6
6
6.8
6
4 = 5.75
12
3
5
2
6
6
7
7
11
8
5
8
10
4
7
0
4
6
10
5
3
5
9
6
2
4
0
6
3
3
8
3
14
8
10
2
1
9
5
1
7
5
7
11
5
3
5
9
8
4
5
9
4
3
5
9
2
4
12
12
1
4
10
7
7
11
7
6
6
7.4
7 5.4
5.6
3
7.2
5.4
6.2
6
5
6.4
6.8
7.4 5.2
mx = 8
_
sx = 4/sqrt(16)
_
8
What is a sampling distribution
of the mean?
_
x
7
8
8
9
2
6
6
6
6
6
6
6
7
7
11
8
5
8
10
4
7
0
4
6
10
5
3
5
9
6
2
4
0
6
3
3
8
3
14
8
10
2
1
9
5
1
7
5
7
11
5
3
5
9
8
4
5
9
4
3
5
9
2
4
12
12
1
4
10
7
mx = 8
_
sx = 4/sqrt(16)
_
8
7
11
7
6
6
4 = 5.75
12 = 5.63
3 =…
5 =…
2… = …
What is a sampling distribution
of the mean?
_
x
7
8
8
9
2
6
6
6
6
6
6
6
7
7
11
8
5
8
10
4
7
0
4
6
10
5
3
5
9
6
2
4
0
6
3
3
8
3
14
8
10
2
1
9
5
1
7
5
7
11
5
3
5
9
8
4
5
9
4
3
5
9
2
4
12
12
1
4
10
7
mx = 8
_
sx = 4/sqrt(16)
_
8
7
11
7
6
6
4 = 5.75
12 = 5.63
3 =…
5 =…
2… = …
Sampling and Variation
Sample Size & Estimated Nail Biting
at closing time by Stock Brokers
0.09
0.08
Frequency
0.07
0.06
n=1000
n=100
n=10
0.05
0.04
0.03
0.02
0.01
0
0
1
2
3
4
5
6
7
Nail Biting
8
9
10
11
12
13
It has been documented that in the general population, the average number
of hours slept a night by Americans is 8, with a standard deviation of 4
hours. I collect sample data on a set of 16 college students, and determine
that their average number of hours sleeping per night is 6 hours (s=1). Do
college students on average sleep less than the general population?
In other words, I am comparing a sample mean (my 16 students) with a
population mean and asking the question, are they different?
Sampling distribution (the distribution of means)
_
mx = 8
_
sx = 4/√16
sample
x=6
s=1
8
6
The z-test
• When comparing a sample mean to a population mean
AND THE POPULATION STANDARD DEVIATION IS
KNOWN, use a z-test.
zobt 
x  mx
sx
Introducing the Z-test
Standard error of the mean and z-score
s
sx =
n
z-score for a sample mean
x mx
Zobt =
sx
State your hypotheses:
H1: College students sleep less than people in the general population.
H0: College students DO NOT sleep less than people in the general
population.
Calculate the critical value:
We need to find the zcrit for a one-tailed test (alpha = .05).
Using the z-table, we can find: zcrit = -1.65
Calculate
_ the obtained statistic.
zobt = (x-mx) /sx
zobt = (6-8)/1 = -2
Make a decision.
Reject the null. College students do indeed sleep less than people
in the general population.
Another z-test example:
In the population, individuals spend on average $10/day on
alcohol, with a standard deviation of $2. I suspect that among
graduate students, the average amount spent is greater than the
population average. I gather together 25 graduate students from
this program and calculate the average amount spent on
alcohol/day: $11.00 (s= $1). Use an alpha level of .05.
How can I test whether or not graduate students differ
significantly from the general population in this regard?
How do I know I need to be using a z-test?
1. Identify your null and alternative hypotheses:
H1: College students spend more money on alcohol per day than a person from
the general population.
H0: College students DO NOT spend more money on alcohol per day than a
person from the general population.
2.
Determine the critical region of the sampling distribution in which
you will reject the null hypothesis:
Since we are using a z-test, the z-distribution is our sampling distribution.
In order to find Zcrit, we must know about the number of tails of our hypothesis, as
well as the alpha level. Alpha = .05, Tails = 1.
Zcrit = 1.65 (This means that if our Zobt value exceeds 1.65, only then will we reject
the null).
3.
Calculate the test statistic:
_
We know that:
4.
Zobt = x – mx
sx
Zobt =
Make a decision about your null hypothesis:
11 - 10
=
2.5
.40
Reject the null.
Yet another z-test example:
In a population of American graduate students, individuals earn on
average $7.25/hour, with a standard deviation of $5. I want to
know whether or not graduate students from Brooklyn earn a
different wage than grad students in general. I gather together 9
graduate students from this program and calculate the average
amount they earn an hour: $11.00 (s= $.50). Use an alpha level of
.01.
How can I test whether or not graduate students differ
significantly from the general population in this regard?
How do I know I need to be using a z-test?
1. Identify your null and alternative hypotheses:
H1: Graduate students in Brooklyn earn a different amount than graduate students
in general.
H0: Graduate students in Brooklyn DO NOT earn a different amount than
graduate students in general.
2.
Determine the critical region of the sampling distribution in which
you will reject the null hypothesis:
Since we are using a z-test, the z-distribution is our sampling distribution.
In order to find Zcrit, we must know about the number of tails of our hypothesis, as
well as the alpha level. Alpha = .01, Tails = 2.
Zcrit = -2.58 and 2.58 (This means that if our Zobt value exceeds either -2.58 or
2.58, only then will we reject the null).
3.
Calculate the test statistic:
_
We know that:
4.
Zobt = x – mx
sx
Zobt = 11 - 7.25 = 2.25
1.67
Make a decision about your null hypothesis: Retain the null.
What do these problems we have been working on have in
common?
In a population of American graduate students, individuals earn on
average $7.25/hour on alcohol, with a standard deviation of $5. I
jokingly ask whether or not graduate students from Brooklyn earn
a different wage than grad students in general. I gather together 9
graduate students from this program and calculate the average
amount they earn an hour: $11.00 (s= $.50). Use an alpha level of
.01.
How do I know I need to be using a z-test?
1. We are comparing a sample mean to a KNOWN population mean.
2. We KNOW the population s.