10.1: Confidence Intervals – The Basics

Review Question!!! If the mean and the standard deviation of a continuous random variable that is normally distributed are 26 and 6 respectively, find an interval that contains 95% of the distribution. A) ( 14, 38) B) ( 8, 44) C) ( 20, 32) D) ( 6, 46)

Review Question!!! A wholesale distributor has found that the amount of a customer’s order is a normal random variable with a mean of $200 and a standard deviation of $50. The distributer takes a sample of 25 orders, what is the mean of the sampling distribution? A) 50 B) 200 C) 250 D)10

Review Question!!! A wholesale distributor has found that the amount of a customer’s order is a normal random variable with a mean of $200 and a standard deviation of $50. The distributer takes a sample of 25 orders, what is the standard deviation of the sampling distribution? A) 50 B) 200 C) 250 D)10

Review Question!!! A wholesale distributor has found that the amount of a customer’s order is a normal random variable with a mean of $200 and a standard deviation of $50. What is the probability that a sample of 25 orders is within $20 of the mean? A) B) 0.31 C) D) 0.5

Introduction Is caffeine dependence real? What proportion of college students engage in binge drinking? How do we answer these questions? Statistical inference provides methods for drawing conclusions about a population from sample data.

Ex 1: IQ and Admissions

Ex 2: Estimation in Pictures.

Ex 3: IQ Conclusion

Confidence Interval & Level 25 samples from the same population gave these 95% confidence intervals. In the long run, 95% of all samples given an interval that contains the population mean μ.

Conditions for Constructing a Confidence Interval for μ

Ex 4: Finding z (Using Table A)

Most Common Confidence Levels Confidence Level Tail AreaZ* 90% 95% 99%

Critical Values

Confidence Interval for a Population Mean (σ Known) When choosing an SRS from a population (having unknown μ and known σ), the level C confidence interval for μ is: = Estimate ± Margin of Error = Estimate ± (Critical Value of z) (Standard Error)

Ex 5: Video Screen Tension A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. Careful study has shown that when the process is operating properly, the standard deviation of the tension readings is σ = 43 mV. Here are the tension readings from an SRS of 20 screens from a single day’s production:

Ex 5: Video Screen Tension

Step 2: Plan Cont.: SRS? We are told that the sample was an SRS. Normality? The sample size is too small to use the central limit theorem (n=20), but the boxplot of the sample data is approximately Normal (no severe skew or outliers) We can assume that the sampling distribution is approximately Normal. Independence? We must assume that at least (10)(20) = 200 video terminals were produces on this day.

Ex 5: Video Screen Tension Step 4: Interpretation: Interpret your results in the context of the problem. We are 90% confident that the true mean tension in the entire batch of video terminals produced that day is between and mV.

How Confidence Intervals Behave We select the confidence interval, and the margin of error follows… We strive for HIGH confidence and a SMALL margin of error. HIGH confidence says that our method almost always gives correct answers. SMALL margin of error says that we have pinned down the parameter quite precisely.

How Confidence Intervals Behave Consider margin of error… The margin of error gets smaller when… z gets smaller. To accept a smaller margin of error, you must be willing to accept lower confidence. σ gets smaller. The standard deviation σ measures the variation in the population. n gets larger. We must take four times as many observations in order to cut the margin of error in half.

Sample Size for a Desired Margin of Error To determine the sample size that will yield a confidence interval for a population mean with a specified margin of error, set the expression for the margin of error to be less than or equal to m and solve for n:

Ex 2: How Many Monkeys? Researchers would like to estimate the mean cholesterol level μ of a particular variety of monkey that is often used in lab experiments. They would like their estimate to be within 1 mg/dl of the true value of μ at a 95% confidence level. A previous study indicated that σ = 5 mg/dl. Obtaining monkeys is time- consuming and expensive, so researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate. We must round up!!! We need 97 monkeys to estimate the cholesterol levels to our satisfaction.

Some Cautions Read the “Cautions” on p

BIG CAUTION We are 90% confident that the true mean tension in the entire batch of video terminals produced that day is between and mV. These number were calculated by a method that give correct results in 95% of all possible samples. We cannot say that the probability is 90% that the true mean falls between and No randomness remains after we draw one particular sample and get from it one particular interval. The true mean either is or is not between and The probability calculation of statistical inference describes how often the METHOD gives correct answers.

ExpressPoll 2/7/ :27:03 AM

ExpressPoll 2/7/ :29:51 AM

ExpressPoll 2/7/ :31:30 AM

ExpressPoll 2/7/ :33:11 AM