Upload
miles-johnson
View
215
Download
0
Embed Size (px)
Citation preview
Confidence Interval for a Mean
when you have a “small” sample...
As long as you have a “large” sample….
A confidence interval for a population mean is:
n
sZx
where the average, standard deviation, and n depend on the sample, and Z depends on the confidence level.
Example
Random sample of 59 students spent an average of $273.20 on Spring 1998 textbooks. Sample standard deviation was $94.40.
09.2420.27359
4.9496.120.273
We can be 95% confident that the average amount spent by all students was between $249.11 and $297.29.
What happens if you can only take a “small” sample?
• Random sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.
• What is the average amount all students slept last night?
If you have a “small” sample...
Replace the Z value with a t value to get:
n
stx
where “t” comes from Student’s t distribution, and depends on the sample size through the degrees of freedom “n-1”.
Student’s t distribution versus Normal Z distribution
-5 0 5
0.0
0.1
0.2
0.3
0.4
Value
dens
ity
T-distribution and Standard Normal Z distribution
T with 5 d.f.
Z distribution
T distribution
• Shaped like standard normal distribution (symmetric around 0, bell-shaped).
• But, t depends on the degrees of freedom “n-1”.
• And, more likely to get extreme t values than extreme Z values.
Graphical Comparison of T and Z Multipliers
0.90 0.92 0.94 0.96 0.98 1.00
0
1
2
3
4
5
Cumulative Probability
Z o
r T
Mul
tiplie
r T with 5 df
Z distribution
Tabular Comparison of T and Z Multipliers
Confidencelevel
t value with5 d.f
Z value
90% 2.015 1.65
95% 2.571 1.96
99% 4.032 2.58
For small samples, T value is larger than Z value.
So,T interval is made to be longer than Z interval.
Back to our example!
Sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.
55.04.615
1145.24.6
n
stx
Need t with n-1 = 15-1 = 14 d.f. For 95% confidence, t14 = 2.145
That is...
We can be 95% confident that average amount slept last night by all students is between 5.85 and 6.95 hours.
Hmmm! Adults need 8 hours of sleep each night.
Logical conclusion:On average, students need more sleep.
(Just don’t get it in this class!)
T-Interval for Mean in Minitab
T Confidence Intervals
Variable N Mean StDev SE Mean 95.0 % CIComb 89 2.011 1.563 0.166 (1.682, 2.340)
We can be 95% confident that the average number of times a “Stat-250-like” student combs his/her is between 1.7 and 2.3 times a day.
T- interval in Minitab
• Select Stat.
• Select Basic Statistics.
• Select 1-Sample t…
• Select desired variable.
• Specify desired confidence level.
• Say OK.
What happens as sample gets larger?
-5 0 5
0.0
0.1
0.2
0.3
0.4
Value
dens
ity
T-distribution and Standard Normal Z distribution
Z distribution
T with 60 d.f.
What happens to CI as sample gets larger?
n
sZx
n
stx
For large samples:
Z and t values become almost identical, so CIs will be almost identical.
Example
Random sample of 64 students spent an average of 3.8 hours on homework last night with a sample standard deviation of 3.1 hours.
Z Confidence Intervals The assumed sigma = 3.10
Variable N Mean StDev 95.0 % CIHomework 64 3.797 3.100 (3.037, 4.556)
T Confidence IntervalsVariable N Mean StDev 95.0 % CIHomework 64 3.797 3.100 (3.022, 4.571)
One not-so-small problem!
• It is only OK to use the t interval for small samples if your original measurements are normally distributed.
• We’ll learn how to check for normality.
Strategy
• If you have a large sample of, say, 30 or more measurements, then don’t worry about normality, and calculate a t-interval.
• If you have a small sample and your data are normally distributed, then calculate a t-interval.
• If you have a small sample and your data are not normally distributed, then stay tuned.