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10-2 Estimating a Population Mean (σ Unknown) Warm-up Question •High School students who take the SAT Mathematics exam a second time generally score higher than on their first try. The change in the score has a Normal distribution with standard deviation σ=50. A random sample of 250 students gain on average x-bar=22 points on their second try. •Construct a 95% Confidence interval for μ •How large a sample of high school students would be needed to estimate the mean change in SAT score μ to within ± 2 points with 95% confidence? Confidence Intervals Involving Z Using the Calculator •Because we don’t know σ, we estimate it by using the sample standard deviation, s. •We then estimate the standard deviation of xbar using s / sqrt(n) •This is called the standard error of the sample mean x-bar “Standard error”: You are estimating the standard deviation…but there will likely be some error involved because we are estimating it from sample data. In other words… the standard error is (most likely) an inaccurate estimate of a (population) standard deviation. The t distributions When we substitute the standard error of xbar ( s/sqrt(n) )for its standard deviation ( σ/sqrt(n)), the distribution of the resulting statistic, t, is not Normal. We call it the t distribution. The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland ("Student" was his pen name). Gosset devised the t-test as a way to cheaply monitor the quality of stout. The t distributions There is a different t-distribution for each sample size n. We specify a t distribution by giving its degrees of freedom, which is equal to n-1 We will write the t distribution with k degrees of freedom as t(k) for short. We also will refer to the standard Normal distribution as the z-distribution. Comparing t and z distributions Compare the shape, center, and spread of the t-distribution with the z-distribution. As the degrees of freedom k increase, the t(k) density curve approaches the N(0,1) curve ever more closely. As the sample size increases, s estimates σ ever more closely. Finding t with Table C Suppose you want to construct a 95% confidence interval for the mean μ of a population based on a SRS of size n=12. What critical value t should you use? Finding t with Table C Suppose you want to construct a 90% confidence interval for the mean μ of a population based on a SRS of size n=15. What critical value t should you use? Suppose you want to construct a 80% confidence interval for the mean μ of a population based on a SRS of size n=5. What critical value t should you use? Our formula is the same as it was for zintervals EXCEPT we replace sigma with s!!! One sample t interval for mu Recall the inference tool-box One sample t interval for mu Environmentalists, government officials, and vehicle manufacturers are all interested in studying the auto exhaust emissions produced by motor vehicles. The table gives the nitrogen oxide (NOX) levels for a random sample of light-duty engines of the same type. One sample t interval for mu Construct a 95% confidence interval for the mean amount of NOX emitted by light-duty engines of this type. One sample t interval for mu Step 1: Parameter Step 2: Conditions Step 3:Calculations One sample t interval for mu Step 1: Parameter Step 2: Conditions Step 3:Calculations Step 4:Interpretation Remember the three C's! Conclusion, Connection, Context One sample t interval for mu Note: When the actual df does not appear in Table C, use the greatest df available that is less than your desired df. Ronald McDonald’s sister Diana Rhea is the purchasing manager for domestic hamburger outlets. The company has decided to provide a free package of Tums to any complaining customer. In order to estimate monthly demand, she took a sample of 5 outlets and found the number of Tums distributed to customers in a month was 250, 280, 220, 280, 320 (a)Find the sample mean and sample standard deviation (b)Construct a 90% confidence interval on the average monthly demand per outlet. Homework!