1. t Distribution t distribution with ∞ degrees of freedom
The t-distribution is used when n is small and s is unknown.
t distribution with ∞ degrees of freedom
Standard normal
distribution
t distribution with 20 degrees of freedom
The t distribution is a family of similar probability distributions that is no different from the standard normal distribution when the sample size is large.
A specific t distribution depends on a parameter known as the degrees of freedom.
Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.
t distribution with 10 degrees of freedom

2. t distribution with ∞ degrees of freedom
If n is large there is no difference between the standard normal and t distributions.
t distribution with ∞ degrees of freedom
Standard normal
distribution
.0250
t.0250 = ?

3. t Distribution
If n is large there is no difference between the standard normal and t distributions.
P( > 1.96) = .0250 t

4. t distribution with ∞ degrees of freedom
If n is large there is no difference between the standard normal and t distributions.
P( > 1.96) = .0250 t
t distribution with ∞ degrees of freedom
Standard normal
distribution
.0250
1.96

5. t Distribution
If n is large there is no difference between the standard normal and t distributions.
P( > 1.96) = .0250 t z Z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
-2.2
.0139
.0136
.0132
.0129
.0125
.0122
.0119
.0116
.0113
.0110
-2.1
.0179
.0174
.0170
.0166
.0162
.0158
.0154
.0150
.0146
.0143
-2.0
.0228
.0222
.0217
.0212
.0207
.0202
.0197
.0192
.0188
.0183
-1.9
.0287
.0281
.0274
.0268
.0262
.0256
.0250
.0244
.0239
.0233
-1.8
.0359
.0351
.0344
.0336
.0329
.0322
.0314
.0307
.0301
.0294
-1.7
.0446
.0436
.0427
.0418
.0409
.0401
.0392
.0384
.0375
.0367

6. t Distribution
If n is large there is no difference between the standard normal and t distributions.

7. t distribution with ∞ degrees of freedom
If n is large there is no difference between the standard normal and t distributions.
t distribution with ∞ degrees of freedom
Standard normal
distribution
.0250
1.96

8. t Distribution with df = 20 t.0250 = ?
If n is large there is no difference between the standard normal and t distributions.
with df = 20
.0250
t.0250 = ?

9. t Distribution with df = 20
If n is large there is no difference between the standard normal and t distributions.
with df = 20
P( > 2.086) = .0250 t

10. t Distribution with df = 20 2.086
If n is large there is no difference between the standard normal and t distributions.
with df = 20
.0250
2.086

11. t Distribution with df = 10 t.0250 = ?
If n is large there is no difference between the standard normal and t distributions.
with df = 10
.0250
t.0250 = ?

12. t Distribution with df = 10
If n is large there is no difference between the standard normal and t distributions.
with df = 10

13. t Distribution with df = 10 2.228
If n is large there is no difference between the standard normal and t distributions.
with df = 10
.0250
2.228

14. Interval Estimate of a Population Mean
The 1 -  confidence interval for unknown m
z/2 is the z value providing an area of /2 in the
upper tail of the standard normal distribution
s is the known population standard deviation
and
n > 2 if the data is normally distributed
n > 30 if the data is roughly symmetric
n > 50 if the data is heavily skewed OR OR

15. Interval Estimate of a Population Proportion
The 1 -  confidence interval for unknown p
z/2 is the z value providing an area of /2 in the
upper tail of the standard normal distribution
and
n > 5/p
n > 5/(1 – p)
and

16. Interval Estimate of a Population Mean
The 1 -  confidence interval for unknown m
t/2 is the t value providing an area of /2 in the
upper tail of the t-distribution
s estimates s because it is unknown
and
n > 2 if the data is normally distributed
n > 15 if the data is roughly symmetric
n > 30 if the data is slightly skewed
n > 50 if the data is heavily skewed OR OR OR

17. Interval Estimate of a Population
Mean
Example: Discount Sounds
Discount Sounds has 260 retail outlets throughout the United States. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. n x
A sample of size of 36 was taken; the sample mean income is $31,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is .95. s
1 - a

18. Interval Estimate of a Population
Mean
Example: Discount Sounds
The margin of error is:
Thus, at 95% confidence, the margin of error is $1,470.
Interval estimate of  is:
$31, $1,470
$29,630 to $32,570
We are 95% confident that the interval contains m.

19. Interval Estimate of a Population
Proportion
Example: Political Science, Inc.
Political Science, Inc. (PSI) specializes in voter polls and
surveys designed to keep political office seekers informed
of their position in a race. Using telephone surveys, PSI interviewers ask registered voters who they would vote for if the election were held that day.
In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor McSame. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor McSame. n
1 - a

20. Interval Estimate of a Population
Proportion
Example: Political Science, Inc.
where: n = 500,
= 220/500 = .44,
z.025 = 1.96 =
PSI is 95% confident that the proportion of all voters
that favor McSame is between and

21. Interval Estimate of a Population
Mean
Example: Apartment Rents
A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 16 efficiency apartments within a half-mile of campus resulted in a sample mean of $650 per month and a sample standard deviation of $55. n x s
1 - a
Compute a 95% confidence interval estimate of the mean rent per month for the population of efficiency apartments within a half-mile of campus. We will assume this population to be normally distributed.

22. Interval Estimate of a Population
Mean
At 95% confidence,
 = .05,
/2 = .025.
t.025 is based on df = n - 1 = = 15

23. Interval Estimate of a Population Mean
Est. Margin
of Error
We are 95% confident that the mean rent per month
for the population of efficiency apartments within a
half-mile of campus is between $ and $
Point Estimate of the population mean
Interval Estimate of the population mean
Interval Estimate of the population mean

24. Sufficient Sample Size for the Interval Estimate of a Population Mean
Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location.
Suppose that Discount Sounds’ management team
wants an estimate of the population mean such that
there is a .95 probability that the margin of error is $500
or less. How large should the sample be?
1-a n? E
At 95% confidence, z.025 = 1.96.
Recall that = 4,500.

25. Sufficient Sample Size for the Interval Estimate of a Population Proportion
Example: Political Science, Inc.
1-a
Suppose that PSI would like a .99 probability that the sample proportion is within .03 of the population proportion. How large should the sample be to meet the required precision? Recall that in the previous example, a sample of 500 similar units yielded a sample proportion of .44. E p*
At 99% confidence,
a = .01,
a/2 = .005,
z.005 = 2.575
Using the sample proportion from a previous sample of the same or similar units
Select a preliminary sample to compute a “preliminary” sample proportion.
If no information is available about p, then .5 is often assumed because it provides the highest possible sample size.

26. Repeated Sampling
s = 10
n = 64
a = 0.05
z.025 = 1.96
.025
.950 

27. Repeated Sampling
s = 10
n = 64
a = 0.05
z.025 = 1.96
.025
.950 

28. Repeated Sampling Example: Political Science, Inc. s = 10 n = 64
z.025 = 1.96
.025
.950 