Confidence Intervals

1
Confidence Intervals
Chapter 6
2
6.2

• Confidence Intervals for the Mean (Small
  Samples)

3
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.
Properties of the t-distribution

• The t-distribution is bell shaped and symmetric
  about the mean.
• 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 ? 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.
4
The t-Distribution

1. The total area under a t-curve is 1 or 100.
2. The mean, median, and mode of the t-distribution
   are equal to zero.
3. 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.
5
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 0.50 0.80 0.90 0.95 0.98
One tail, ? 0.25 0.10 0.05 0.025 0.01
d.f. Two tails, ? 0.50 0.20 0.10 0.05 0.02
1 1.000 3.078 6.314 12.706 31.821
2 .816 1.886 2.920 4.303 6.965
3 .765 1.638 2.353 3.182 4.541
4 .741 1.533 2.132 2.776 3.747
5 .727 1.476 2.015 2.571 3.365
d.f. n 1 5 1 4
tc 2.776
c 0.95
Continued.
6
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.
7
Confidence Intervals and t-Distributions
Constructing a Confidence Interval for the Mean
t-Distribution
In Words In Symbols

1. Identify the sample statistics n, ?, 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.

d.f. n 1
Left endpoint ??E Right endpoint ? E Interval
??E lt µ lt ? E
8
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.
? 95
s 21
n 20
tc 1.729
d.f. 19
86.9 lt µ lt 103.1
? E 95 8.1
We are 90 confident that the mean wait time for
all customers is between 86.9 and 103.1 seconds.
9
Normal or t-Distribution?
Is n ? 30?
Is the population normally, or approximately
normally, distributed?
You cannot use the normal distribution or the
t-distribution.
Is ? known?
10
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 lt 30
and the distribution is skewed.
c.) n 25, the distribution is normal, ? 4.12
The normal distribution would be used because
although n lt 30, the population standard
deviation is known.