1. Section 7.3 Sample Means

2. A look back at proportions… You may have noticed that proportions ALWAYS deal with categorical data. You may have noticed that proportions ALWAYS deal with categorical data. We’ve looked at the proportion of first-year college students who applied to more than one school. We’ve looked at the proportion of first-year college students who applied to more than one school. We’ve looked at the proportion of people who are left handed. We’ve looked at the proportion of people who are left handed.

3. Quantitative Variables When we are interested in quantitative variables, like the income of a household, how long a car lasts, or the blood pressure of a patient, we look at other statistics. When we are interested in quantitative variables, like the income of a household, how long a car lasts, or the blood pressure of a patient, we look at other statistics. One of the most popular statistics for quantitative variables is the sample mean. One of the most popular statistics for quantitative variables is the sample mean.

4. Why Be Mean? Averages are less variable than individual observations. Averages are less variable than individual observations. Which is more likely: finding one person whose IQ is at least 130 or finding a random sample of individuals whose AVERAGE IQ is at least 130? Which is more likely: finding one person whose IQ is at least 130 or finding a random sample of individuals whose AVERAGE IQ is at least 130? In fact, averages are more NORMAL than individual observations. In fact, averages are more NORMAL than individual observations.

5. The Mean and Standard Deviation of x-bar Since x-bar is an unbiased estimator of μ, what should the mean of x-bar be? Since x-bar is an unbiased estimator of μ, what should the mean of x-bar be? What happens to the standard deviation as n increases?

6. Some facts about x-bar X-bar is an unbiased estimator of μ. X-bar is an unbiased estimator of μ. The larger the sample, the smaller the variation of x-bar values. The larger the sample, the smaller the variation of x-bar values.

7. BONUS!!!!! Thank you for checking the PowerPoints. Thank you for checking the PowerPoints. If you write down the word “Joetro” at the top of your test, I will give you 4 bonus points. If you write down the word “Joetro” at the top of your test, I will give you 4 bonus points.

8. Example A grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125 mm. The standard deviation is σ = 0.002 mm. The machine operator inspects a random sample of 4 axles each hour and records the sample mean diameter. What are the mean and standard deviation of the sampling distribution of x-bar? A grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125 mm. The standard deviation is σ = 0.002 mm. The machine operator inspects a random sample of 4 axles each hour and records the sample mean diameter. What are the mean and standard deviation of the sampling distribution of x-bar?

9. Example A grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125 mm. The standard deviation is σ = 0.002 mm. How many axles would you need to sample if you wanted the standard deviation of the sampling distribution of x-bar to be within 0.0005 mm? A grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125 mm. The standard deviation is σ = 0.002 mm. How many axles would you need to sample if you wanted the standard deviation of the sampling distribution of x-bar to be within 0.0005 mm?

10. PCFS for Means Parameter Parameter This time our parameter is µ. This time our parameter is µ. Conditions Conditions Normality: today, the problem will tell you that the population is distributed normally. Normality: today, the problem will tell you that the population is distributed normally. Independence: population ≥ 10n Independence: population ≥ 10n Formula Formula Sentence Sentence

11. Example A grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125 mm. The standard deviation is σ = 0.002 mm. Assume the population of all axles made is distributed normally. A grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125 mm. The standard deviation is σ = 0.002 mm. Assume the population of all axles made is distributed normally. What is the probability that a random sample of 4 axles has a mean diameter of at least 40.1265 mm? What is the probability that a random sample of 4 axles has a mean diameter of at least 40.1265 mm? What is the probability that an individual axle is at least 40.1265 mm in diameter? What is the probability that an individual axle is at least 40.1265 mm in diameter?

12. Don’t you have any other examples? Women’s heights are distributed normally with a mean of 64.5 inches and a standard deviation of 2.5 inches. Women’s heights are distributed normally with a mean of 64.5 inches and a standard deviation of 2.5 inches. What is the probability that a randomly selected young woman is taller than 66.5 inches? What is the probability that a randomly selected young woman is taller than 66.5 inches? What is the probability that the mean height of a sample of 10 young women is greater than 66.5 inches? What is the probability that the mean height of a sample of 10 young women is greater than 66.5 inches?