1. Normal Distributions and Sampling Distributions

2. Types of Distributions
Frequency Distributions
Probability Distributions

3. Normal Distributions Same mean, different standard deviations
Different means

4. X and Z Distributions 100 200 X 2.0 Z
P= from minus infinity to X=200 and Z=2.0
100
200 X
(μ = 100, σ = 50)
2.0 Z
(μ = 0, σ = 1)

5. Computing Cumulative Probability
Start with an X value
Compute the Z value for X, µ, and δ
Z = (X-µ)/δ
From the table, find the cumulative probability
Start with an X value
Use the NORMDIST function for X, Ẋ, δ, and TRUE to find the cumulative probability
Tabular Method
Normdist Method

6. Probability for a Range of X Values
P(8.0 to 8.6) = Cumulative Probability (8.6)
– Cumulative Probability (8.0) X
8.0
8.6

7. Frequency Distribution v Sampling Distribution
Event is the estimation of the mean (X bar) from a sample of size n. X
Sampling X X
Frequency Distribution with µ
Sampling Distribution for Ẋ

8. Sampling Distribution Statisticsδ X
Population Distribution
Sampling Distribution for µ

9. Confidence Intervals
“Based on a sample of 1,500 households, the percentage of voters in favor of Proposition X is 40%, with a sampling error of plus or minus 3%.”

10. Confidence Interval
Probability that the true population mean will lie within a certain interval around the sampling distribution medium
95% Confidence Interval
Zα/2 = -1.96
Zα/2 = 1.96
Z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
X units:
Point Estimate
Point Estimate

11. Confidence Coefficient,
Confidence Intervals
Confidence Coefficient,
Confidence Level
Zα/2 value
80%
90%
95%
98%
99%
99.8%
99.9%
0.80
0.90
0.95
0.98
0.99
0.998
0.999
1.28
1.645
1.96
2.33
2.58
3.08
3.27

12. Example
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Solution:

13. Confidence Interval Use Normal Distribution With δ Use t Distribution
Intervals
Use
Normal
Distribution
With δ
Population
Mean
Population
Proportion
Use
t Distribution
based on the sample standard deviation S computed from sample instead of δ
σ Known
σ Unknown

14. Student’s t Distribution
Assumptions
Population standard deviation is unknown
Population is normally distributed
If population is not normal, use large sample
Use Student’s t Distribution
Confidence Interval Estimate:
(where tα/2 is the critical value of the t distribution with n -1 degrees of freedom and an area of α/2 in each tail)

15. Degrees of Freedom
Idea: Number of observations that are free to vary after sample mean has been calculated
Example: Suppose the mean of 3 numbers is 8.0
If the mean of these three values is 8.0,
then X3 must be 9
(i.e., X3 is not free to vary)
X1 = 7
X2 = 8
X3 = ?
Here, the sample size (n) = 3
So degrees of freedom = n – 1 = 3 – 1 = 2

16. Degrees of Freedom d.f. = n-1
For confidence intervals based on sample standard deviations,
d.f. = n-1
Where n is the sample size
© 2011 Pearson Education, Inc.  Publishing as Prentice Hall

17. Student’s t Distribution
Note: t Z as n increases so (n n)
Standard Normal
(t with df = ∞)
t (df = 13)
t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
t (df = 5) t

18. Student’s t Table .05 2 t /2 = 0.05 2.920 Upper Tail Area df .25 .10
Let: n = df = n - 1 =  = /2 = 0.05 df
.25
.10
.05 1
1.000
3.078
6.314 2
0.817
1.886
2.920
/2 = 0.05 3
0.765
1.638
2.353
The body of the table contains t values, not probabilities t
2.920

19. Example t Distribution Values
Confidence Level
t (10 d.f.)
t (20 d.f.)
t (30 d.f.) z
.90
1.812
1.725
1.697
1.645
.95
2.228
2.086
2.042
1.96
.99
3.169
2.845
2.750
2.58
As sample size n increases, df (n-1) increases.
As df increases, t approaches z
So at large sample sizes, t and z are the same

20. Example of the t distribution
A random sample of n = 25 has X = 50 and
S = 8. Form a 95% confidence interval for μ
d.f. = n – 1 = 24, so
The confidence interval is
≤ μ ≤

21. Using the Excel TINV Distribution
TINV(Probability, df)
For a 95% confidence level and sample size of 11
df is 10 (n-1)
P = 95%
Probability is .05 (Confidence level is 1-P)
Equation is = TINV(.05,10)
Value is (same as in previous table)

22. Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Based on a sample of 70, 95% of our faculty members have PhDs.
σ Known
σ Unknown

23. Confidence Intervals for the Population Proportion, π
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:

24. Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where
Zα/2 is the standard normal value for the level of confidence desired
p is the sample proportion
n is the sample size
Note: must have np > 5 and n(1-p) > 5

25. Example
A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left- handers.

26. Determining the Required Sample Size
for a desired error size and confidence level

27. Determining Sample Size
(continued)
To determine the required sample size for the mean, you must know:
The desired level of confidence (1 - ), which determines the critical value, Zα/2
The acceptable sampling error, e
The standard deviation, σ

28. Required Sample Size Example
If  = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?
So the required sample size is n = 220
(Always round up)

29. If σ is unknown
If unknown, σ can be estimated when using the required sample size formula
Use a value for σ that is expected to be at least as large as the true σ
Select a pilot sample and estimate σ with the sample standard deviation, S

30. Ethical Issues
A confidence interval estimate (reflecting sampling error) should always be included when reporting a point estimate
The level of confidence should always be reported
The sample size should be reported
An interpretation of the confidence interval estimate should also be provided