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+ Z-Interval for µ So, the formula for a Confidence Interval for a population mean is xz * n To be honest, σ is never known. So, this formula isn’t used very much. The only homework problem regarding this formula is #55, a problem where you are asked to find the sample size for a desired margin of error for a CI for µ. Let’s practice that first. + Finding n for a desired MOE for a CI for μ High school students who take the SAT Math exam a second time generally score higher than on their first try. Past data suggest that the score increase has a standard deviation of 50 points. How large a sample of high school students would be needed to estimate the mean change in SAT score to within 2 points with 95% confidence? Standard Error We don’t know σ. Therefore, we will estimate σ based on s, the standard deviation of the sample. σ is estimated from the data, the result is called the standard error of the statistic. When s The standard error of x is . n Finding the standard error of the mean (like #59 in tonight’s homework) Standard error of the mean is often abbreviated SEM. Suppose that a SRS of 20 people yields an average IQ of 107 with a standard deviation of 14.8. What is the SEM? + Characteristics of the t-distributions They are similar to the normal distribution. They are symmetric, bell-shaped, and are centered around 0. The t-distributions have more spread than a normal distribution. They have more area in the tails and less in the center than the normal distribution. That’s because using s to estimate σ introduces more variation. As the degrees of freedom increase, the tdistribution more closely resembles the normal curve. As n increases, s becomes a better estimator of σ. + Degrees of freedom Unlike the z-statistic, the shape of the t-distribution changes based on the sample size. t(k) stands for a t-distribution with k degrees of freedom. So, if our sample size is 30, and σ is unknown, the distribution is a tdistribution with 29 degrees of freedom. This would be written t(29). + The t-distribution How many degrees of freedom are there if n=3? If n=10? What happens to the tdistribution as n increases? What do I mean by n=∞ (normal)? + Using Table B Now that we are studying a different distribution, we’ll use a different table. Table B gives the t-distribution critical values. Let’s look for the t* value for an upper tail area of .05 with n = 15. What is the t* value for 18 degrees of freedom with probability 0.90 to the left of t*? Suppose you want to construct a 95% confidence interval for the mean of a population based on a SRS of size n = 12. What critical value t* should you use? Table B to Find Critical t* Values Upper-tail probability p df .05 .025 .02 .01 10 1.812 2.228 2.359 2.764 11 1.796 2.201 2.328 2.718 12 1.782 2.179 2.303 2.681 z* 1.645 1.960 2.054 2.326 90% 95% 96% 98% Confidence level C In Table B, we consult the row corresponding to df = n – 1 = 11. We move across that row to the entry that is directly above 95% confidence level. The desired critical value is t * = 2.201. Estimating a Population Mean Suppose you want to construct a 95% confidence interval for the mean µ of a Normal population based on an SRS of size n = 12. What critical t* should you use? + Using t Interval for a Population Mean Conditions The One-Sample for Inference t Interval about for aaPopulation PopulationMean Mean •Random: Choose an The SRSdata of size come n from fromaapopulation random sample havingofunknown size n from mean theµ.population A level C confidence of interest orinterval a randomized for µ is experiment.s x t* x • Normal: The population has a Normal distribution or the sample size is large n (n ≥ 30). where t* is the critical value for the tn – 1 distribution. • Independent: The method for calculating a confidence interval assumes that Use this interval only when: individual observations are independent. To keep the calculations accurate wheniswe sample replacement from(na ≥finite (1) reasonably the population distribution Normal orwithout the sample size is large 30), population, we should check the 10% condition: verify that the sample size (2) the at least 10population times as large is nopopulation more thanis1/10 of the size.as the sample. Estimating a Population Mean The one-sample t interval for a population mean is similar in both reasoning and computational detail to the one-sample z interval for a population proportion. As before, we have to verify three important conditions before we estimate a population mean. + One-Sample Video Screen Tension Read the Example Parameter: µ = the true mean tension of all the video terminals produced this day Conditions: Random: We are told that the data come from a random sample of 20 screens from the population of all screens produced that day. Normal: Since the sample size is small (n < 30), we must check whether it’s reasonable to believe that the population distribution is Normal. Examine the distribution of the sample data. These graphs give no reason to doubt the Normality of the population Independent: Because we are sampling without replacement, we must check the 10% condition: we must assume that at least 10(20) = 200 video terminals were produced this day. Estimating a Population Mean on page 508. + Example: Video Screen Tension DO: Using our calculator, we find that the mean and standard deviation of the 20 screens in the sample are: x 306.32 mV .10 sx 36.21 mV .05 .025 Since n = 20, we use the t distribution with df = 19 to find the critical value. From Table B, we find t* = 1.729. Upper-tail probability p df and 18 1.130 1.734 2.101 19 1.328 1.729 2.093 20 1.325 1.725 2.086 90% 95% 96% Confidence level C Estimating a Population Mean Example on page 508. We want to estimate the true mean tension µ of all the video terminals produced this day at a 90% confidence level. + Example: Therefore, the 90% confidence interval for µ is: sx 36.21 x t* 306.32 1.729 n 20 306.32 14 (292.32, 320.32) CONCLUDE: We are 90% confident that the interval from 292.32 to 320.32 mV captures the true mean tension in the entire batch of video terminals produced that day. t Procedures Wisely Using One-Sample t Procedures: The Normal Condition • Sample size less than 15: Use t procedures if the data appear close to Normal (roughly symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t. • Sample size at least 15: The t procedures can be used except in the presence of outliers or strong skewness. • Large samples: The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 30. Estimating a Population Mean Except in the case of small samples, the condition that the data come from a random sample or randomized experiment is more important than the condition that the population distribution is Normal. Here are practical guidelines for the Normal condition when performing inference about a population mean. + Using + Confidence Intervals – puzzles for math nerds statistic critical value standard deviation of statistic Parameter Mean Proportion Statistic x p p Standard Deviation of the Statistic n p(1 p) n Standard Error of the Statistic s n p(1 p) n