•

Module 11: Confidence Intervals Part 1
B. Creating confidence intervals to estimate the population mean with small samples (n 30).
The formula for estimating a population mean for a small sample size (n
30) and
unknown:
1.
Population Size Unknown:
±
t
/2, n-1
Where
is the point estimate for the population mean and
t
/2, n-1
is the confidence interval half width
Note:
t
/2, n-1
will be explained on the next slide.
2.
Population Size Known:
±
t
/2, n-1
Where
is the point estimate for the population mean and
t
/2, n-1
is the confidence interval half width
Note: The formulas for estimating a population with a small sample size are very similar to those for a large sample size.
The only difference is that
is replaced with
t
/2, n-1
•

Module 11: Confidence Intervals Part 1
B. Creating confidence intervals to estimate the population mean with small samples (n 30).
With smaller sample sizes or if the distribution is unknown, we use the
t
statistic.
Unlike the Z-Statistic, which we could
look up on slide 7 for most common confidence levels, the t-statistic must be calculated (using Excel) for each individual
problem.
There are two variables with the t statistic:
n - 1 (also called "df" or degrees of freedom) and , which is the same as for the z-distribution, but excel calculates it
differently. The way Excel does it can be kind of confusing and illogical so just do exactly what I do and you will be fine.
Some examples:
•
Confidence
α
α/2
sample size (n)
n-1
t
α/2, n-1
t
α/2, n-1
Excel Syntax
0.99
0.01
0.005
25
24
t
.005, 24
2.7969
=TINV(2*0.005,24)
0.99
0.01
0.005
20
19
t
.005, 19
2.8609
=TINV(2*0.005,19)
0.98
0.02
0.01
25
24
t
.01, 24
2.4922
=TINV(2*0.01,24)
0.98
0.02
0.01
20
19
t
.01, 19
2.5395
=TINV(2*0.01,19)
0.95
0.05
0.025
25
24
t
.025 24
2.0639
=TINV(2*0.025,24)
0.95
0.05
0.025
20
19
t
.025, 19
2.0930
=TINV(2*0.025,19)
0.90
0.10
0.05
25
24
t
.05 24
1.7109
=TINV(2*0.05,24)
0.90
0.10
0.05
20
19
t
.05, 19
1.7291
=TINV(2*0.05,19)

Module 11: Confidence Intervals Part 1
B. Creating confidence intervals to estimate the population mean with small samples (n 30).
Example 5
:
A researcher wants to estimate the average amount of comp time accumulated per week in the aerospace
industry. He randomly samples 18 managers and measures the amount if extra time they work during a specific week and
obtains the results in the table below. Use this data to construct a 90% confidence interval for the average amount of comp
time worked per week by managers in the aerospace industry.
First we use Excel to calculate the sample mean ()and sample standard deviation(s).
= 13.5556, s = 7.8006, n = 18, and n - 1 = 17
With a 90% level of confidence, α/2 = .05
Next we use Excel to calculate
t
/2, n-1
=
t
•
6
8
20
9
8
15
3
17
11
0
25
29
21
12
7
21
16
16
13.5556
=AVERAGE(A1:A18)
7.8006
=STDEV(A1:A18)
t
α/2, n-1
t
.05, 17
1.7396
=TINV(2*0.05,17)

Module 11: Confidence Intervals Part 1
B. Creating confidence intervals to estimate the population mean with small samples (n 30).
Example 5
:
From the previous slide, we have:
= 13.5556, s = 7.8006, n = 18, and n - 1 = 17, and =
t
.05, 17
=
1.7396
To construct our confidence, we just plug the data into our formula:
±
t
/2, n-1
= 13.5556 ± 1.7396
= 13.5556 ± 3.1985
P(10.3571
16.7541) = 90%
Interpretation:
We're 90% confident that the mean amount of weekly comp time for managers in the aerospace industry is
between 10.3571 and 16.7541 hours.

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- Spring '14
- DebraACasto
- Statistics, Normal Distribution, Standard Deviation