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.
Random sample of 59 students spent an average of $273.20 on Spring 1998 textbooks. Sample standard deviation was $94.40. Example 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: where “t” comes from Student’s t distribution, and depends on the sample size through the degrees of freedom “n-1”.
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.
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, 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 % CI Comb 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 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 64 3.797 3.100 (3.037, 4.556) T Confidence Intervals Variable N Mean StDev 95.0 % CI Homework 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.