Transcript The Mean and Standard Deviation

The Population Mean and
Standard Deviation
σ
μ
X
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Computing the Mean and the
Standard Deviation in Excel
• μ = AVERAGE(range)
• δ = STDEV(range)
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Exercise
• Compute the mean, standard deviation, and
variance for the following data:
• 1 2 3 3 4 8 10
• Check Figures
– Mean = 4.428571
– Standard deviation = 3.309438
– Variance = 10.95238
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The Normal Distribution
P(-∞ to X)
μ
X
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Solving for P(-∞ to X) in Excel
• P(-∞ to X) =
• NORMDIST(X, mean, stdev, cumulative)
– X = value for which we want P(-∞ to X)
– Mean = µ
– Stdev = δ
– Cumulative = True (It just is)
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Exercise in Solving for P(-∞ to X)
• What portion of the adult population is under
6 feet tall if the mean for the population is 5
feet and the standard deviation is 1 foot?
– Check figure = 0.841345
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P(X to ∞)
P(X to ∞)
μ
X
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P(X to ∞)
• P(X to ∞) = 1 – P(-∞ to X)
P(-∞ to X)
P(X to ∞)
μ
P=1.0
X
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Exercise
• What portion of the adult population is OVER
6 feet tall if the mean for the population is 5
feet and the standard deviation is 1 foot?
– Check figure = 0.158655
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P(X1 to X2)
P(X1 < X < X2)
X1
X2
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P(X1 to X2) in Excel
• P(X1 to X2) = P(-∞ to X2) - P(-∞ to X1)
• P(X1 to X2)=NORMDIST(X2…)–NORMDIST(X1…)
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Exercise in P(X1 to X2) in Excel
• What portion of the adult population is
between 6 and 7 feet tall if the mean for the
population is 5 feet and the standard
deviation is 1 foot?
– Check figure = 0.135905
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Computing X
P(-∞ to X)
μ
X
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Computing X in Excel
• X = NORMINV(probability, mean, stdev)
– Probability is P(-∞ to X)
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Exercise in Computing X in Excel
• An adult population has a mean of 5 feet and
a standard deviation is 1 foot. Seventy-five
percent of the people are shorter than what
height?
– Check figure = 5.67449
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Z Distribution
• A transformation of normal distributions into
a standard form with a mean of 0 and a
standard deviation of 1. It is sometimes useful.
μ=8
σ = 10
8 8.6
P(X < 8.6)
μ=0
σ=1
X
0 0.12
Z
P(Z < 0.12)
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Computing P(-∞ to Z) in Excel
• Z = (X-μ)/δ
• P(-∞ to Z) = NORMDIST(Z, mean, stdev,
cumulative)
(X  )
Z
– Mean = 0
– Stdev = 1

– Z = (X-μ)/δ
– Cumulative = True (It just is)
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Exercise in Computing P(-∞ to Z) in Excel
• An adult population has a mean of 5 feet and a
standard deviation is 1 foot. Compute the Z value for
4.5 feet all. What portion of all people are under 4.5
feet tall
– Z check figure = -.5 (the minus is important)
– P check figure = 0.308537539
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Z Distribution
• A transformation of normal distributions into
a standard form with a mean of 0 and a
standard deviation of 1. It is sometimes useful.
μ=8
σ = 10
8 8.6
P(X < 8.6)
μ=0
σ=1
X
0 0.12
Z
P(Z < 0.12)
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Computing Z in Excel
• Z for a certain value of P(-∞ to Z)
=NORMINV(probilility, mean, stdev)
– Probability = P(-∞ to Z)
– Mean = 0
– Stdev = 1
• Change the Z value to an X value if necessary
– Z = (X-μ)/δ, so
–X=µ+Zδ
X  μ  Zσ
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Exercise in Computing Z in Excel
• An adult population has a mean of 5 feet and
a standard deviation is 1 foot. 25% of the
population is greater than what height?
– Check figure for Z = 0.67449
– Check figure for X = 0.308537539
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Sampling Distribution of the Mean
Normal
Population
Distribution
μ
Normal
Sampling
Distribution
(has the same
mean)
x
δ is the
Population
Standard
Deviation
δXbar is the
Sample
Standard
Deviation.
μx
x
δXbar = δ/√n
δXbar << δ
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Sampling Distribution of the Mean
• For the sampling distribution of the mean.
– The mean of the sampling distribution is Xbar
– The standard deviation of the sampling
distribution of the mean, δXbar, is δ/√n
• This only works if δ is known, of course.
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Exercise in Using Excel in the Sampling
Distribution of the Mean
• The sample mean is 7. The population
standard distribution is 3. The sample size is
100
• Compute the probability that the true mean is
less than 5.
• Compute the probability that the true mean is
3 to 5
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Confidence Interval if δ is Known
• Using X
1  α  0.95 so α  0.05
α
α
 0.025
2
X units:
 0.025
2
Lower
Confidence
Limit
Xmin
Point
Estimate
for Xbar
Upper
Confidence
Limit
Xmax
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Confidence Interval
• 95% confidence level
• Xmin is for P(-∞ to Xmin) = 0.025
• Xmax is for P(-∞ to Xmax) = 0.975
• X = NORMINV(probability, mean, stddev)
– Here, stdev is δXbar = δ/√n
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Exercise
• For a sample of 25, the sample mean is 100.
The population standard deviation is 50.
• What is the standard deviation of the
sampling distribution?
– Check figure: 10
• What are the limits of the 95% confidence
level?
– Check figure for minimum: 80.40036015
– Check figure for maximum: 119.5996
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Confidence Interval if δ is Known
• Done Using Z
1  α  0.95 so α  0.05
α
α
 0.025
2
Z units:
 0.025
2
Zα/2 = -1.96
0
Zα/2 = 1.96
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Confidence Intervals with Z in Excel
• Xmin = Xbar – Zα/2 * δ/√n
– Why?
– Because multiplying a Z value by δ/√n gives the X
value associated with the Z value
• Xmax = Xbar + Zα/2 * δ/√n
• Common Zα/2 value:
– 95% confidence level = 1.96
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Exercise in Confidence Intervals
with Z in Excel
• The sampling mean Xbar is 100. The population
standard deviation, δ, is 50. The sample size is
25. What are Xmin and Xmax for the 95%
confidence level?
– Check figure: Zα/2 = 1.96
– Xmin = 80.4 (same as before)
– Xmax = 119.6 (same as before)
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Confidence Intervals, δ Unknown
• Use the sample standard deviation S instead
of δXbar.
– No need to divide S by the square root of n
– Because S is not based on the population δ
• Use the t distribution instead of the normal
distribution.
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Computing the t values
• Z = TINV(probability, df)
– probability is P(-∞ to X)
– df = degrees of freedom = n-1 for the sampling
distribution of the mean.
• Xmin = Xbar – Z(.025,n-1)*S
• Xmax = Xbar + Z(.975,n-1)*S
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Exercise
• For a sample of 25, the sample mean is 100. The
sample standard deviation is 5.
• What is Z for the 95% confidence interval?
– Check figure 2.390949
• What is the lower X limit?
– Check figure 88.04525 (With δ known, was
80.40036015)
• What is the upper X limit?
– Check figure 111.9547 (With δ known, was 119.5996)
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t test for two samples
• What is the probability that two samples have
the same mean?
Sample Mean
Sample A
1
3
Sample B
1
2
5
5
7
5
4
8
9
10
5.714286
9
10
5.571429
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The t Test Analysis
• Go to the
Data tab
• Click on
data
analysis
• Select t-Test
for TwoSample(s)
with Equal
Variance
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With Our Data and .05 Confidence Level
t stat = 0.08
t critical for twotail (H1 = not
equal) = 2.18.
T stat < t Critical,
so do not reject
the null
hypothesis of
equal means.
Also, α is 0.94,
which is far larger
than .05
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t Test:
Two-Sample, Equal Variance
• If the variances of the two samples are
believed to be the same, use this option.
• It is the strongest t test—most likely to reject
the null hypothesis of equality if the means
really are different.
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t Test:
Two-Sample, Unequal Variance
• Does not require equal variances
– Use if you know they are unequal
– Use is you do not feel that you should assume
equality
• You lose some discriminatory power
– Slightly less likely to reject the null hypothesis of
equality if it is true
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t Test:
Two-Sample, Paired
• In the sampling, the each value in one
distribution is paired with a value in the other
distribution on some basis.
• For example, equal ability on some skill.
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z Test for Two Sample Means
• Population
standard
deviation
is
unknown.
• Must
compute
the
sample
variances.
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z test
• Data tab
• Data
analysis
• z test
sample
for two
means
Z value is greater than z Critical for two tails (not equal),
so reject the null hypothesis of the means being equal.
Also, α = 2.31109E-08 < .05, so reject.
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Exercise
• Repeat the analysis above.
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