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AP Statistics Section 10.2 A CI for Population Mean When is Unknown In Section 10.1B, we constructed a confidence interval for the population mean when we knew the population standard deviation . It is extremely unlikely that we would actually know the population standard deviation, however. In this section, we will discover how to construct a confidence interval for an unknown population mean when we don’t know the standard deviation . We will do this by estimating from the data. This need to estimate with s introduces additional error into our calculations. To account for this, * we will use a critical value of t instead of z* when computing our confidence interval. Note the following properties of a t distribution: The density curves of the t distributions are similar in shape to the standard Normal, or z, distribution (i.e. bell - shaped and centered at zero Unlike the standard Normal distribution, there is a different t distribution for each sample size n. We specify a particular t distribution by giving its __________________ degrees of freedom ( _____ df ). When we perform inference about using a t distribution, the appropriate degrees of freedom is equal to ______. n - 1 We will write the t distribution with k degrees of freedom as _____. t(k) The spread of the t distributions is slightly greater than that of the z distribution. The t distributions are less concentrated around the mean and have more probability in the tails. This is what accounts for the increased error in using s instead of . As the degrees of freedom increase, the t curve approaches the standard Normal curve ever more closely. This happens because s approximates more accurately as the sample size increases. Table B, gives the values of t* for various degrees of freedom and various upper-tail probabilities. When the actual degrees of freedom does not appear in Table B, use the largest degrees of freedom that is less than your desired degrees of freedom. Example: Determine the appropriate value of t* for a confidence interval for with the given confidence level and sample size. a) 98% with n = 22 .01 .98 .01 df 22 1 21 t 2.518 Example: Determine the appropriate value of t* for a confidence interval for with the given confidence level and sample size. b) 90% with n = 38 .05 .90 .05 df 38 1 37 must use df 30 on the table t 1.697 TI 84: 2 nd VARS DISTR invt ENTER invt (area to left, df) As before, we need to verify three important conditions before we estimate a population mean. SRS: Our data are a SRS of size n from the population of interest or come from a randomized experiment. This condition is very important. Normality of x : The population has a Normal distribution or : n 15 Use t procedures if sample data appears roughly Normal. : n 15 The t procedures can be used except in the presence of outliers or strong skewness in the sample data. The t procedures are robust. : n 30 The t procedures can be used even for clearly skewed distributions. However, outliers are still a concern. You may still refer to the Central Limit Theorem in this situation. Independence: The method for calculating a confidence interval assumes that individual observations are independent. To keep the calculations reasonably accurate when we sample without replacement from a finite population, we should verify that the population size is at least _______________________(________). 10 times the sample size N 10n x t s n Example: A number of groups are interested in studying the auto exhaust emissions produced by motor vehicles. Here is the amount of nitrogen oxides (NOX) emitted by light-duty engines (grams/mile) from a random sample of size n = 46. Construct and interpret a 95% confidence interval for the mean amount of NOX emitted by light-duty engines of this type. Parameter: The population of interest is ____________________. light - duty engines We want to estimate , the ____________________________. mean amount of NOX emitted Conditions: Since we do not know , use ______________________ a one - sample t interval SRS: Data comes from a random sample of size n 46. Normality of x : With n 46, the CLT applies and the distribution of x is approximately Normal. Independence: Since sampling w/o replacement, we must assume the population is at least 460 so that N 10n. Calculation: x 1.329 s .484 n 46 df 46 1 45 t 2.021 .484 1.329 2.021 46 1.329 .144 (1.185,1.473) Interpretation: We are 95% confident that the mean of NOX emissions for light - duty engines is between 1.185 and 1.473 grams/mile. TI 83/84: STATS Tests T - Interval Standard Error When the standard deviation of a statistic, i.e. x or p̂ , is estimated from the data, the result is called the standard error of the statistic. Some textbooks simply refer to standard error as the standard deviation of the sampling distribution, x or p̂ , whether it is estimated from the data or not.