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Unformatted text preview: Chapter 9 Section 1 The Logic in Constructing Confidence Intervals about a Population Mean where the Population Standard Deviation is Known Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 1 of 39 Confidence Intervals ● Learning objectives Compute a point estimate of the population mean 2 Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known) 3 Understand the role of margin of error in constructing a confidence interval 4 Determine the sample size necessary for estimating the population mean within a specified margin of error 1 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 2 of 39 Confidence Intervals ● Learning objectives Compute a point estimate of the population mean 2 Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known) 3 Understand the role of margin of error in constructing a confidence interval 4 Determine the sample size necessary for estimating the population mean within a specified margin of error 1 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 3 of 39 Chapter 9 – Section 1 ● The environment of our problem is that we want to estimate the value of an unknown population mean ● The process that we use is called estimation ● This is one of the most common goals of statistics Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 4 of 39 Chapter 9 – Section 1 ● Estimation involves two steps Step 1 – to obtain a specific numeric estimate, this is called the point estimate Step 2 – to quantify the accuracy and precision of the point estimate ● The first step is relatively easy ● The second step is why we need statistics Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 5 of 39 Chapter 9 – Section 1 ● Some examples of point estimates are The sample mean to estimate the population mean The sample standard deviation to estimate the population standard deviation The sample proportion to estimate the population proportion The sample median to estimate the population median Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 6 of 39 Confidence Intervals ● Learning objectives Compute a point estimate of the population mean 2 Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known) 3 Understand the role of margin of error in constructing a confidence interval 4 Determine the sample size necessary for estimating the population mean within a specified margin of error 1 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 7 of 39 Chapter 9 – Section 1 ● The most obvious point estimate for the population mean is the sample mean ● Now we will use the material in Chapter 8 on the sample mean to quantify the accuracy and precision of this point estimate Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 8 of 39 Chapter 9 – Section 1 ● An example of what we want to quantify We want to estimate the miles per gallon for a certain car We test some number of cars We calculate the sample mean … it is 27 27 miles per gallon would be our best guess Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 9 of 39 Chapter 9 – Section 1 ● How sure are we that the gas economy is 27 and not 28.1, or 25.2? ● We would like to make a statement such as “We think that the mileage is 27 mpg and we’re pretty sure that we’re not too far off” Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 10 of 39 Chapter 9 – Section 1 ● A confidence interval for an unknown parameter is an interval of numbers Compare this to a point estimate which is just one number, not an interval of numbers ● The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 11 of 39 Chapter 9 – Section 1 ● What does the level of confidence represent? ● If we have a process for calculating confidence intervals with a 90% level of confidence Assume that we know the population mean We then obtain a series of 50 random samples We apply our process to the data from each random sample to obtain a confidence interval for each ● Then, we would expect that 90% of those 50 confidence intervals (or about 45) would contain our population mean Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 12 of 39 Chapter 9 – Section 1 ● If we expect that a method would create intervals that contain the population mean 90% of the time, we call those intervals 90% confidence intervals ● If we have a method for intervals that contain the population mean 95% of the time, those are 95% confidence intervals ● And so forth Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 13 of 39 Chapter 9 – Section 1 ● The level of confidence is always expressed as a percent ● The level of confidence is described by a parameter α ● The level of confidence is (1 – α) • 100% When α = .05, then (1 – α) = .95, and we have a 95% level of confidence When α = .01, then (1 – α) = .99, and we have a 99% level of confidence Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 14 of 39 Chapter 9 – Section 1 ● To tie the definitions together (in English) We are using the sample mean to estimate the population mean With each specific sample, we can construct a 95% confidence interval As we take repeated samples, we expect that 95% of these intervals would contain the population mean Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 15 of 39 Chapter 9 – Section 1 ● To tie all the definitions together (using statistical terms) We are using a point estimator to estimate the population mean We wish to construct a confidence interval with parameter α, the level of confidence is (1 – α) • 100% As we take repeated samples, we expect that (1 – α) • 100% of the resulting intervals will contain the population mean Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 16 of 39 Chapter 9 – Section 1 ● Back to our 27 miles per gallon car “We think that the mileage is 27 mpg and we’re pretty sure that we’re not too far off” ● Putting in numbers, “We estimate the gas mileage is 27 mpg and we are 90% confident that the real mileage of this model of car is between 25 and 29 miles per gallon” Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 17 of 39 Chapter 9 – Section 1 “We estimate the gas mileage is 27 mpg” ● This is our point estimate “and we are 90% confident that” ● Our confidence level is 90% (i.e. (i.e. αα == 0.10) 0.10) “the real mileage of this model of car” ● The population mean “is between 25 and 29 miles per gallon” ● Our confidence interval is (25, 29) Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 18 of 39 Chapter 9 – Section 1 ● In this section, we assume that we know the standard deviation of the population (σ) ● This is not very realistic … but we need it for right now ● We’ll solve this problem in a better way (where we don’t know what σ is) in the next section … but first we’ll do this one Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 19 of 39 Chapter 9 – Section 1 ● If n is large enough, i.e. n ≥ 30, we can assume that the sample means have a normal distribution with standard deviation σ / √ n ● We look up a standard normal calculation 95% of the values in a standard normal are between – 1.96 and 1.96 … in other words between ± 1.96 ● We now use this applied to a general normal calculation Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 20 of 39 Chapter 9 – Section 1 ● The values of a general normal random variable are less than ± 1.96 times its standard deviation away from its mean 95% of the time ● Thus the sample mean is within ± 1.96 σ / √ n of the population mean 95% of the time Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 21 of 39 Chapter 9 – Section 1 ● Because the sample mean has an approximately normal distribution, it is in the interval 1.96 n around the (unknown) population mean 95% of the time ● We can flip that around to solve for the population mean μ Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 22 of 39 Chapter 9 – Section 1 ● After we solve solve for for the the population populationmean meanμ,μ,we wefind find that μ is within within the the interval interval x 1.96 n around the the (known) (known) sample samplemean mean“95% “95%ofofthe the time” ● This isn’t exactly true in the mathematical sense as the population mean is not a random variable … that’s why we call this a “confidence” instead of a “probability” Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 23 of 39 Chapter 9 – Section 1 ● Thus a 95% confidence interval for the mean is x 1.96 n ● This is in the form Point estimate ± margin of error ● The margin of error here is 1.96 • σ / √ n Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 24 of 39 Chapter 9 – Section 1 ● For our car mileage example Assume that the sample mean was 27 mpg Assume that we tested 40 cars Assume that we knew that the population standard deviation was 6 mpg ● Then our 95% confidence interval would be 6 27 1.96 40 or 27 ± 1.9 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 25 of 39 Chapter 9 – Section 1 ● If we wanted to compute a 90% confidence interval, or a 99% confidence interval, etc., we would just need to find the right standard normal value ● Frequently used confidence levels, and their critical values, are 90% corresponds to 1.645 95% corresponds to 1.960 99% corresponds to 2.575 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 26 of 39 Chapter 9 – Section 1 ● The numbers 1.645, 1.960, and 2.575 are written as Z-values z0.05 = 1.645 … P(Z ≥ 1.645) = .05 z0.025 = 1.960 … P(Z ≥ 1.960) = .025 z0.005 = 2.575 … P(Z ≥ 2.575) = .005 where Z is a standard normal random variable Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 27 of 39 Chapter 9 – Section 1 ● Why do we use α = 0.025 for 95% confidence? ● To be within something 95% of the time We can be too low 2.5% of the time We can be too high 2.5% of the time ● Thus the 5% confidence that we don’t have is split as 2.5% being too high and 2.5% being too low … needing α = 0.025 (or 2.5%) Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 28 of 39 Chapter 9 – Section 1 ● In general, for a (1 – α) • 100% confidence interval, we need to find zα/2, the critical Z-value ● zα/2 is the value such that P(Z ≥ zα/2) = α/2 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 29 of 39 Chapter 9 – Section 1 ● Once we know these critical values for the normal distribution, then we can construct confidence intervals for the sample mean x z / 2 n to x z / 2 n Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 30 of 39 Chapter 9 – Section 1 ● Learning objectives Compute a point estimate of the population mean 2 Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known) 3 Understand the role of margin of error in constructing a confidence interval 4 Determine the sample size necessary for estimating the population mean within a specified margin of error 1 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 31 of 39 Chapter 9 – Section 1 ● If we write the confidence interval as 27 ± 2 then we would call the number 2 (after the ±) the margin of error ● So we have three ways of writing confidence intervals (25, 29) 27 ± 2 27 with a margin of error of 2 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 32 of 39 Chapter 9 – Section 1 ● The margin of error depends on three factors The level of confidence (α) The sample size (n) The standard deviation of the population (σ) ● We’ll now calculate the margin of error ● Once we know the margin of error, we can state the confidence interval Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 33 of 39 Chapter 9 – Section 1 ● The margin of errors would be 1.645 • σ / √ n for 90% confidence intervals 1.960 • σ / √ n for 95% confidence intervals 2.575 • σ / √ n for 99% confidence intervals Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 34 of 39 Chapter 9 – Section 1 ● Learning objectives Compute a point estimate of the population mean 2 Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known) 3 Understand the role of margin of error in constructing a confidence interval 4 Determine the sample size necessary for estimating the population mean within a specified margin of error 1 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 35 of 39 Chapter 9 – Section 1 ● Often we have the reverse problem where we want an experiment to result in an answer with a particular accuracy ● We have a target margin of error ● We need to find the sample size (n) needed to achieve this goal Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 36 of 39 Chapter 9 – Section 1 ● For our car miles per gallon, we had σ = 6 ● If we wanted our margin of error to be 1 for a 95% confidence interval, then we would need 6 1.96 1.96 1 n n ● Solving for n would get us n = (1.96 • 6)2 or that n = 138 cars would be needed Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 37 of 39 Chapter 9 – Section 1 ● We can write this as a formula ● The sample size n needed to result in a margin of error E for (1 – α) • 100% confidence is n z / 2 E 2 ● Usually we don’t get an integer for n, so we would need to take the next higher number (the one lower wouldn’t be large enough) Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 38 of 39 Summary: Chapter 9 – Section 1 ● We can construct a confidence interval around a point estimator if we know the population standard deviation σ ● The margin of error is calculated using σ, the sample size n, and the appropriate Z-value ● We can also calculate the sample size needed to obtain a target margin of error Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 39 of 39 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