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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: s x Z n 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. 94.4 273.20 1.96 273.20 24.09 59 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: s x t n 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 and Standard Normal Z distribution 0.4 Z distribution density 0.3 0.2 T with 5 d.f. 0.1 0.0 -5 0 Value 5 T distribution • Very similar to standard normal distribution, except: • t depends on the degrees of freedom “n-1” • more likely to get extreme t values than extreme Z values Let’s compare t and Z values Confidence t value with Z value level 5 d.f 2.015 1.65 90% 2.571 1.96 95% 4.032 2.58 99% For small samples, T value is larger than Z value. So,T interval is made to be longer than Z interval. OK, enough theorizing! Let’s get 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 s 1 x t 6.4 2.145 6.4 0.55 n 15 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: Students need more sleep. (Just don’t get it in this class!) What happens as sample gets larger? T-distribution and Standard Normal Z distribution 0.4 Z distribution density 0.3 T with 60 d.f. 0.2 0.1 0.0 -5 0 Value 5 What happens to CI as sample gets larger? x Z x t s n s n For large samples: Z and t values become almost identical, so CIs are 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 Homework N Mean 64 3.797 StDev 3.100 95.0 % CI (3.037, 4.556) T Confidence Intervals Variable N Mean Homework 64 3.797 StDev 3.100 95.0 % CI (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 in a minute. Strategy for deciding how to analyze • If you have a large sample of, say, 60 or more measurements, then don’t worry about normality, and use the t-interval. • If you have a small sample and your data are normally distributed, then use the tinterval. • If you have a small sample and your data are not normally distributed, then do not use the t-interval, and stay tuned for what to do. Using Minitab, let’s go practice this strategy … Z interval in Minitab • Find sample standard deviation by Stat>> Basic Statistics >> Display Descriptive Statistics • Ask for Z interval by Stat >> Basic Statistics >> 1-Sample Z…. • Select desired variable. • Type sample stand. deviation in Sigma box. • Specify desired confidence level. OK. t interval in Minitab • Ask for t interval by Stat >> Basic Statistics >> 1-Sample t…. • Select desired variable. • Specify desired confidence level. • Say OK.