Stated or CLT (large sample size) Where we are… Proportions Means Mean Standard Deviation Random? SRS? 10% condition Normal? Stated or CLT (large sample size) Inference Use z ? conditions

8.3: Estimating a Population Mean Intro: The Gettysburg Address 8.3: Estimating a Population Mean

The Gettysburg Address Four score and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation or any nation so conceived and so dedicated can long endure. We are met on a great battlefield of that war. We have come to dedicate a portion of that field as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this. But in a larger sense, we cannot dedicate, we cannot consecrate, we cannot hallow this ground. The brave men, living and dead who struggled here have consecrated it far above our poor power to add or detract. The world will little note nor long remember what we say here, but it can never forget what they did here. It is for us the living rather to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great talk remaining before us – that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion – that we here highly resolve that these dead shall not have died in vain, that this nation under God shall have a new birth of freedom, and that government of the people, by the people, for the people shall not perish from the earth.

The Gettysburg Address Suppose you are interested in determining the mean word length of Lincoln’s Gettysburg Address. Surely such an important speech uses long, sophisticated words Work with your neighbor to determine a method for selecting a sample of 30 words. (Note: there are 268 words in the speech.) Work with your neighbor to find the mean word length from your sample, 𝑥 . (For consistency, use the random numbers I generated in my calculator.)

The Gettysburg Address Use word number: 4 5 8 12 21 23 31 32 54 55 65 67 81 84 101 107 161 163 166 170 187 188 195 225 229 248 254 264 265 268 Find the mean word length for this sample. 𝑥 =4.4333

The Gettysburg Address What is the point estimate for the mean word length? How will we come up with an estimate that we are more comfortable with? estimate ±(critical value)∙(standard dev.) 𝑥 ±𝑧*∙ 𝜎 𝑛

Suppose 𝜎 is known. Can we assume our sampling distribution is Normal? If 𝜎=2.1233, find a 95% confidence interval. 𝑥 ±𝑧* 𝜎 𝑛 4.4333±1.96∙ 2.1233 30 (3.6735, 5.1931)

Finding a Sample Size. To determine the sample size 𝑛 that will yield a level C confidence interval for a population mean with a specified margin of error ME: 𝑧*∙ 𝜎 𝑛 ≤𝑀𝐸

Finding a Sample Size. Researchers would like to estimate the mean cholesterol level µ of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1 milligram per deciliter (mg/dl) of the true value of µ at a 95% confidence level. A previous study involving this variety of monkey suggests that the standard deviation of cholesterol level is about 5 mg/dl. How many monkeys do they need for their sample?

But what if 𝜎 is unknown? 𝑥 ±𝑧* 𝜎 𝑛 𝑥 ±(critical value)∙ ? 𝑛 𝑥 ±𝑧* 𝜎 𝑛 𝑥 ±(critical value)∙ ? 𝑛 We are now using two statistics to find our interval, rather than just one. As a result, our distribution no longer follows the 𝑧-distribution, but rather the 𝑡-distribution. 𝑠

𝑧 vs. 𝑡 𝑧= 𝑥 −𝜇 𝜎 𝑛 𝑡 𝑡 like 𝑧 Standardized value, like 𝑧. 𝑧= 𝑥 −𝜇 𝜎 𝑛 𝑡 like 𝑧 Standardized value, like 𝑧. Tells us how many standard deviations away 𝑥 is from the mean. 𝑡 𝑠 𝑛

Degrees of Freedom (df) When we describe a t distribution we must identify its degrees of freedom because there is a different t statistic for each sample size. The degrees of freedom for the one-sample t statistic is (n – 1). t(k) = t-distribution with k degrees of freedom.

Consider z vs. t Graph What happens as the sample size increases? Y1: normalpdf(x) – the standard normal curve 2nd / Vars / 1:normalpdf( Y2: tpdf (x, 1) 2nd / Vars / 4:tpdf( What happens as the sample size increases? Graph Window Xmin = -4 Xmax = 4 Xscl = 1 Ymin = 0 Ymax = .4 Yscl = .1

The t Distributions; Degrees of Freedom When comparing the density curves of the standard Normal distribution and t distributions, several facts are apparent: The density curves of the t distributions are similar in shape to the standard Normal curve. The spread of the t distributions is a bit greater than that of the standard Normal distribution. The t distributions have more probability in the tails and less in the center than does the standard Normal. As the degrees of freedom increase, the t density curve approaches the standard Normal curve ever more closely. We can use Table B in the back of the book to determine critical values t* for t distributions with different degrees of freedom.

Upper-tail probability p The desired critical value is t * = 2.201. Using Table B to Find Critical t* Values 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? In Table B, we consult the row corresponding to df = n – 1 = 11. 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 We move across that row to the entry that is directly above 95% confidence level. The desired critical value is t * = 2.201.

Using Table B What critical value t* would you use for a t distribution with 18 degrees of freedom having probability .90 to the left of t*? Suppose you wanted to construct a 95% confidence interval based on an SRS of size n =12. What critical value t* should you use? Why is z* on this table?

Finally…Constructing a Confidence Interval State: Define the population parameter () Plan: Identify the appropriate inference method. Check conditions (random, 10%, normal) Do: Perform the calculations ( 𝑥 ±𝑡*∙ 𝑠 𝑛 ) Conclude: Interpret your interval in the context of the problem.

When you have stats (computer output): Auto Pollution Environmentalists, government officials, and vehicle manufacturers are all interested in studying the auto exhaust emissions produced by motor vehicles. The major pollutants in auto exhaust from gasoline engines are hydrocarbons, carbon monoxide, and nitrogen oxides (NOX). Researches collected data on the NOX levels (in grams/mile) for a random sample of 41 light-duty engines of the same type. The mean NOX reading was 1.3083 and the standard deviation was 0.4201. Construct a 99% confidence interval for the mean amount of NOX emitted by light-duty engines of this type. The EPA sets a limit of 1.0 gram/mile for NOX emissions. Are you convinced that this type of engine has a mean NOX level of 1.0 or less? Use your interval to support your answer.

When you have data instead of stats: Video Screen Tension Suppose the manufacturer of video terminals wants to test screen tension. Here are the tension readings from an SRS of 20 screens: Construct and interpret a 90% confidence interval for the mean tension, , of all the screens produced on this day. 269.5 297 269.6 283.3 304.8 280.4 233.5 257.4 317.5 327.4 264.7 307.7 310 343.3 328.1 342.6 338.8 340.1 374.6 336.1

Video

Back to the Gettysburg Address Use what we have learned in this section to estimate the mean word length, , in the Gettysburg Address. (Use a 95% confidence interval.)

Using t Procedures Wisely The stated confidence level of a one-sample t interval for µ is exactly correct when the population distribution is exactly Normal. No population of real data is exactly Normal. The usefulness of the t procedures in practice therefore depends on how strongly they are affected by lack of Normality. Definition: An inference procedure is called robust if the probability calculations involved in the procedure remain fairly accurate when a condition for using the procedures is violated. Fortunately, the t procedures are quite robust against non-Normality of the population except when outliers or strong skewness are present. Larger samples improve the accuracy of critical values from the t distributions when the population is not Normal.

Homework Pg. 518: #49 – 52, 55 – 59 odd, 63 – 67 odd, 71, 73, 75 – 78 all