The Population Mean and
Standard Deviation
μ
σ
X
1

Computing the Mean and the
Standard Deviation in Excel
• μ = AVERAGE(range)
• δ = STDEV(range)
2

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
3

The Normal Distribution
P(-∞ to X)
μ
X
4

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)
5

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
6

P(X to ∞)
P(X to ∞)
μ X
7

P(X to ∞)
• P(X to ∞) = 1 – P(-∞ to X)
P(-∞ to X)
P(X to ∞)
μ
P=1.0
X
8

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
9

P(X
1 to X
2
)
P(X1 < X < X2)
X
1
X
2
10

P(X
1 to X
2
) in Excel
• P(X
1 to X
2
) = P(-∞ to X
2
) - P(-∞ to X
1
)
• P(X
1 to X
2
)=NORMDIST(X
2
…)–NORMDIST(X
1
…)
11

Exercise in P(X
1 to X
2
) 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
12

P(-∞ to X)
Computing X
μ
X
13

Computing X in Excel
• X = NORMINV(probability, mean, stdev)
– Probability is P(-∞ to X)
14

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
15

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
μ = 0
σ = 1
P(X < 8.6)
8 8.6
X
0 0.12
P(Z < 0.12)
Z
16

Computing P(-∞ to Z) in Excel
• Z = (X-μ)/δ
• P(-∞ to Z) = NORMDIST(Z, mean, stdev, cumulative)
– Mean = 0
( X


– Stdev = 1
Z


)
– Z = (X-μ)/δ
– Cumulative = True (It just is)
17

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
18

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
μ = 0
σ = 1
P(X < 8.6)
8 8.6
X
0 0.12
P(Z < 0.12)
Z
19

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 + Z δ
X
 μ 
Z σ
20

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
21

Sampling Distribution of the Mean
Normal
Population
Distribution
Normal
Sampling
Distribution
(has the same mean)
μ
μ x x x
δ is the
Population
Standard
Deviation
δ
Xbar is the
Sample
Standard
Deviation.
δ
Xbar
= δ/√n
δ
Xbar
<< δ
22

Sampling Distribution of the Mean
• For the sampling distribution of the mean.
– The mean of the sampling distribution is X bar
– The standard deviation of the sampling distribution of the mean, δ
Xbar
, is δ/√n
• This only works if δ is known, of course.
23

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
24

Confidence Interval if δ is Known
• Using X
1
 α 
0.95
so
α 
0.05
α
2

0.025
X units:
Lower
Confidence
Limit
Xmin
Point
Estimate for X bar
Upper
Confidence
Limit
Xmax
α
2

0.025
25

Confidence Interval
• 95% confidence level
• X min is for P(-∞ to X min
) = 0.025
• X max is for P(-∞ to X max
) = 0.975
• X = NORMINV(probability, mean, stddev)
– Here, stdev is δ
Xbar
= δ/√n
26

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
27

Confidence Interval if δ is Known
• Done Using Z
1
 α 
0.95
so
α 
0.05
α
2

0.025
Z units:
Z
α/2
= -1.96
0
Z
α/2
= 1.96
α
2

0.025
28

Confidence Intervals with Z in Excel
• X min
= X bar
– Z
α/2
* δ/√n
– Why?
– Because multiplying a Z value by δ/√n gives the X value associated with the Z value
• X max
= X bar
+ Z
α/2
* δ/√n
• Common Z
α/2 value:
– 95% confidence level = 1.96
29

Exercise in Confidence Intervals with Z in Excel
• The sampling mean X bar is 100. The population standard deviation, δ, is 50. The sample size is
25. What are X min confidence level?
and X max for the
95%
– Check figure: Z
α/2
= 1.96
– X min
= 80.4 (same as before)
– X max
= 119.6 (same as before)
30

Confidence Intervals, δ Unknown
• Use the sample standard deviation S instead of δ
Xbar
.
– 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.
31

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.
• X min
= X bar
– Z(.025,n-1)*S
• X max
= X bar
+ Z(.975,n-1)*S
32

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)
33

t test for two samples
• What is the probability that two samples have the same mean?
Sample Mean
Sample A
1
3
5
9
10
5
7
5.714286
Sample B
1
2
5
9
10
4
8
5.571429
34

• Go to the
Data tab
• Click on data analysis
• Select t-Test for Two-
Sample(s) with Equal
Variance
The t Test Analysis
35

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
36

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.
37

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
38

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.
39

z Test for Two Sample Means
• Population standard deviation is unknown.
• Must compute the sample variances.
40

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.
41

Exercise
• Repeat the analysis above.
42