Transcript Chapter 6: Confidence Intervals

Chapter 6
Confidence Intervals
§ 6.2
Confidence Intervals
for the Mean
(Small Samples)
The t-Distribution
When a sample size is less than 30, and the random variable x
is approximately normally distributed, it follow a t-distribution.
t x μ
s
n
Properties of the t-distribution
1. The t-distribution is bell shaped and symmetric about the mean.
2. The t-distribution is a family of curves, each determined by a
parameter called the degrees of freedom. The degrees of freedom
are the number of free choices left after a sample statistic such as x
is calculated. When you use a t-distribution to estimate a
population mean, the degrees of freedom are equal to one less than
the sample size.
d.f. = n – 1
Degrees of freedom
Continued.
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The t-Distribution
3. The total area under a t-curve is 1 or 100%.
4. The mean, median, and mode of the t-distribution are equal to zero.
5. As the degrees of freedom increase, the t-distribution approaches the
normal distribution. After 30 d.f., the t-distribution is very close to
the standard normal z-distribution.
The tails in the t-distribution
are “thicker” than those in the
standard normal distribution.
d.f. = 2
d.f. = 5
0
t
Standard
normal curve
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Critical Values of t
Example:
Find the critical value tc for a 95% confidence when the
sample size is 5.
Appendix B: Table 5: t-Distribution
Level of
confidence, c
One tail, 
d.f. Two tails, 
1
2
3
4
5
0.50
0.25
0.50
1.000
.816
.765
.741
.727
0.80
0.10
0.20
3.078
1.886
1.638
1.533
1.476
0.90
0.95
0.98
0.05
0.025
0.01
0.10
0.05
0.02
6.314 12.706 31.821
2.920 4.303 6.965
2.353 3.182 4.541
2.132 2.776 3.747
2.015 2.571 3.365
d.f. = n – 1 = 5 – 1 = 4
tc = 2.776
c = 0.95
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Continued.
5
Critical Values of t
Example continued:
Find the critical value tc for a 95% confidence when the
sample size is 5.
95% of the area under the t-distribution curve with 4
degrees of freedom lies between t = ±2.776.
c = 0.95
tc =  2.776
tc = 2.776
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Confidence Intervals and t-Distributions
Constructing a Confidence Interval for the Mean:
Distribution
In Words
t-
In Symbols
1. Identify the sample statistics n, x ,
and s.
2. Identify the degrees of freedom,
the level of confidence c, and the
critical value tc.
3. Find the margin of error E.
4. Find the left and right endpoints
and form the confidence interval.
x 
x
n
( x  x )2
s
n 1
d.f. = n – 1
E  tc s
n
Left endpoint: x  E
Right endpoint: x  E
Interval: x  E    x  E
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Constructing a Confidence Interval
Example:
In a random sample of 20 customers at a local fast food
restaurant, the mean waiting time to order is 95 seconds,
and the standard deviation is 21 seconds. Assume the wait
times are normally distributed and construct a 90%
confidence interval for the mean wait time of all customers.
n = 20
x  95
d.f. = 19
tc = 1.729
x  E = 95 ± 8.1
s = 21
21  8.1
E  tc s  1.729 
n
20
86.9 < μ < 103.1
We are 90% confident that the mean wait time for all
customers is between 86.9 and 103.1 seconds.
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Normal or t-Distribution?
Use the normal distribution with
Is n  30?
Yes
E  zc σ .
n
If  is unknown, use s instead.
No
Is the population normally, or
approximately normally,
distributed?
No
Yes
You cannot use the normal
distribution or the t-distribution.
Use the normal distribution with
Is  known?
No
Yes
E  zc σ .
n
Use the t-distribution with
E  tc s
n
and n – 1 degrees of freedom.
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Normal or t-Distribution?
Example:
Determine whether to use the normal distribution, the
t-distribution, or neither.
a.) n = 50, the distribution is skewed, s = 2.5
The normal distribution would be used because the
sample size is 50.
b.) n = 25, the distribution is skewed, s = 52.9
Neither distribution would be used because n < 30 and
the distribution is skewed.
c.) n = 25, the distribution is normal,  = 4.12
The normal distribution would be used because although
n < 30, the population standard deviation is known.
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