The Population Mean and Standard Deviation

1
The Population Mean andStandard Deviation
s
X
µ
2
Computing the Mean and the Standard Deviation in
Excel

• µ AVERAGE(range)
• d STDEV(range)

3
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

4
The Normal Distribution
P(-8 to X)
µ
X
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Solving for P(-8 to X) in Excel

• P(-8 to X)
• NORMDIST(X, mean, stdev, cumulative)
• X value for which we want P(-8 to X)
• Mean µ
• Stdev d
• Cumulative True (It just is)

6
Exercise in Solving for P(-8 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 8)
P(X to 8)
µ
X
8
P(X to 8)

• P(X to 8) 1 P(-8 to X)

P(-8 to X)
P(X to 8)
µ
X
P1.0
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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 lt X lt X2)
X1
X2
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P(X1 to X2) in Excel

• P(X1 to X2) P(-8 to X2) - P(-8 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(-8 to X)
µ
X
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Computing X in Excel

• X NORMINV(probability, mean, stdev)
• Probability is P(-8 to X)

15
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 s 10
µ 0 s 1
Z
X
0.12
0
8.6
8
P(X lt 8.6)
P(Z lt 0.12)
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Computing P(-8 to Z) in Excel

• Z (X-µ)/d
• P(-8 to Z) NORMDIST(Z, mean, stdev, cumulative)
• Mean 0
• Stdev 1
• Z (X-µ)/d
• Cumulative True (It just is)

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Exercise in Computing P(-8 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 s 10
µ 0 s 1
Z
X
0.12
0
8.6
8
P(X lt 8.6)
P(Z lt 0.12)
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Computing Z in Excel

• Z for a certain value of P(-8 to Z)
  NORMINV(probilility, mean, stdev)
• Probability P(-8 to Z)
• Mean 0
• Stdev 1
• Change the Z value to an X value if necessary
• Z (X-µ)/d, so
• X µ Z d

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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
d is the Population Standard Deviation
Normal Sampling Distribution (has the same mean)
dXbar is the Sample Standard Deviation. dXbar
d/vn dXbar ltlt d
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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, dXbar, is d/vn
• This only works if d 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 d is Known

• Using X

PointEstimatefor Xbar
Lower Confidence Limit Xmin
Upper Confidence Limit Xmax
X units
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Confidence Interval

• 95 confidence level
• Xmin is for P(-8 to Xmin) 0.025
• Xmax is for P(-8 to Xmax) 0.975
• X NORMINV(probability, mean, stddev)
• Here, stdev is dXbar d/vn

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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 d is Known

• Done Using Z

Za/2 -1.96
Za/2 1.96
Z units
0
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Confidence Intervals with Z in Excel

• Xmin Xbar Za/2 d/vn
• Why?
• Because multiplying a Z value by d/vn gives the X
  value associated with the Z value
• Xmax Xbar Za/2 d/vn
• Common Za/2 value
• 95 confidence level 1.96

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Exercise in Confidence Intervalswith Z in Excel

• The sampling mean Xbar is 100. The population
  standard deviation, d, is 50. The sample size is
  25. What are Xmin and Xmax for the 95 confidence
  level?
• Check figure Za/2 1.96
• Xmin 80.4 (same as before)
• Xmax 119.6 (same as before)

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Confidence Intervals, d Unknown

• Use the sample standard deviation S instead of
  dXbar.
• No need to divide S by the square root of n
• Because S is not based on the population d
• Use the t distribution instead of the normal
  distribution.

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Computing the t values

• Z TINV(probability, df)
• probability is P(-8 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 d known, was
  80.40036015)
• What is the upper X limit?
• Check figure 111.9547 (With d 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 A Sample B
1 1
3 2
5 5
5 4
7 8
9 9
10 10
Sample Mean 5.714286 5.571429
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The t Test Analysis

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

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With Our Data and .05 Confidence Level
t stat 0.08 t critical for two-tail (H1 not
equal) 2.18. T stat lt t Critical, so do not
reject the null hypothesis of equal means. Also,
a is 0.94, which is far larger than .05
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t TestTwo-Sample, Equal Variance

• If the variances of the two samples are believed
  to be the same, use this option.
• It is the strongest t testmost likely to reject
  the null hypothesis of equality if the means
  really are different.

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t TestTwo-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 TestTwo-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, a 2.31109E-08 lt .05,
so reject.
42
Exercise

• Repeat the analysis above.