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## Inference for a Mean

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**Inference 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. Z* depends on the confidence level.**As long as you have a “large” sample….**A test statistic for a population mean is: where the average, standard deviation, and n depend on the sample. is the value specified in the null.**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.**A sample of 59 students spent an average of $273.20 on**textbooks with a standard deviation of $94.40. Do students spend less than $300 on average? Example There is enough evidence, at 0.05 level, to conclude that, on average, students spend less than $300 on textbooks.**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? • Is the average amount less than 7 hours?**If you have a “small” sample...**Replace the Z multiplier with a t multiplier to get: where “t*” comes from Student t distribution, and depends on the sample size through the degrees of freedom “n-1”.**If you have a “small” sample...**Replace the Z statistic with the t statistic: Again, “t” follows the Student’s t distribution, which depends on the sample size through the degrees of freedom “n-1”.**Student 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 longer than a Z interval, and for a given test statistic the P-value is larger.**Back to our CI 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 or her hair 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.**And to our HT example!**New sample of 18 students slept an average of 6.01 hrs last night with standard deviation of 1.11 hrs. Do students sleep less than 7 hours on average? H0: = 7 vs. HA: < 7 If the population mean is 7, how likely is it that a sample of 18 students would sleep an average as low as 6.01 hours? Or, how likely is it that we’d get a t statistic as low as -3.78?**T-test for Mean in Minitab**T-Test of the Mean Test of mu = 7.000 vs mu < 7.000 Variable N Mean StDev SE Mean T P SleepHrs 18 6.011 1.113 0.262 -3.77 0.0008 If the population mean was 7, it is not likely (P-value = 0.0008) that we’d get a sample mean as small as 6.011 Reject the null hypothesis. There is enough evidence to conclude that students sleep on average less than 7 hours.**T- test in Minitab**• Select Stat. • Select Basic Statistics. • Select 1-Sample t… • Select desired variable. • Specify the null mean in “Test mean” box. • Select the alternative hypothesis. • Say OK.**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)**Example**Random sample of 139 students own an average of 12.7 pairs of shoes with a sample standard deviation of 9.6 pairs. Z-Test Test of mu = 10.000 vs mu > 10.000 The assumed sigma = 9.63 Variable N Mean StDev SE Mean Z P Shoes 139 12.669 9.625 0.816 3.27 0.0006 T-Test of the Mean Test of mu = 10.000 vs mu > 10.000 Variable N Mean StDev SE Mean T P Shoes 139 12.669 9.625 0.816 3.27 0.0007**One not-so-small problem!**• It is only OK to use the t interval for small samples if your original measurements are normally distributed.**Strategy**• If you have a large sample of, say, 30 or more measurements, then don’t worry about normality, and use a t-interval or do a t-test. • If you have a small sample and your data are normally distributed, then use a t-interval or do a t-test. • If you have a small sample and your data are not normally distributed, then use nonparametric hypothesis tests.