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: 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 $ on Spring 1998 textbooks. Sample standard deviation was $ We can be 95% confident that the average amount spent by all students was between $ and $

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: 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

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

Tabular Comparison of T and Z Multipliers 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. Need t with n-1 = 15-1 = 14 d.f. For 95% confidence, t 14 = 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 % CI Comb (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?

What happens to CI as sample gets larger? 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 % CI Homework (3.037, 4.556) T Confidence Intervals Variable N Mean StDev 95.0 % CI Homework (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.