Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 1 of 39 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 2 of 39 Confidence Intervals ●Learning objectives  Compute a point estimate of the population mean  Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known)  Understand the role of margin of error in constructing a confidence interval  Determine the sample size necessary for estimating the population mean within a specified margin of error

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 3 of 39 Confidence Intervals ●Learning objectives  Compute a point estimate of the population mean  Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known)  Understand the role of margin of error in constructing a confidence interval  Determine the sample size necessary for estimating the population mean within a specified margin of error

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 4 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 5 of 39 Chapter 9 – Section 1 ●Estimation involves two steps  Step 1 – to obtain a specific numeric estimate, this is called the point estimate ●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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 6 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 7 of 39 Confidence Intervals ●Learning objectives  Compute a point estimate of the population mean  Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known)  Understand the role of margin of error in constructing a confidence interval  Determine the sample size necessary for estimating the population mean within a specified margin of error

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 8 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 9 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 10 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 11 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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 12 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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 13 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 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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 14 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 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 15 of 39 Chapter 9 – Section 1 ●To tie the definitions together (in English)  We are using the sample mean to estimate the population mean ●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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 16 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 ●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% ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 17 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” ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 18 of 39 Chapter 9 – Section 1 “We estimate the gas mileage is 27 mpg” ●This is our point estimate “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. α = 0.10) “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. α = 0.10) “the real mileage of this model of car” ●The population mean “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. α = 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 19 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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 20 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 21 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 22 of 39 Chapter 9 – Section 1 ●Because the sample mean has an approximately normal distribution, it is in the interval around the (unknown) population mean 95% of the time ●Because the sample mean has an approximately normal distribution, it is in the interval 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 23 of 39 Chapter 9 – Section 1 ●After we solve for the population mean μ, we find that μ is within the interval around the (known) sample mean “95% of the time” ●After we solve for the population mean μ, we find that μ is within the interval around the (known) sample mean “95% of 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 24 of 39 Chapter 9 – Section 1 ●Thus a 95% confidence interval for the mean is ●This is in the form Point estimate ± margin of error ●The margin of error here is 1.96 σ / √ n

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 25 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 ●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 or 27 ± 1.9

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 26 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  95% corresponds to  99% corresponds to 2.575

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 27 of 39 Chapter 9 – Section 1 ●The numbers 1.645, 1.960, and are written as Z-values  z 0.05 = … P(Z ≥ 1.645) =.05  z = … P(Z ≥ 1.960) =.025  z = … P(Z ≥ 2.575) =.005 where Z is a standard normal random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 28 of 39 Chapter 9 – Section 1 ●Why do we use α = 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 ●Why do we use α = 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 = (or 2.5%)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 29 of 39 Chapter 9 – Section 1 ●In general, for a (1 – α) 100% confidence interval, we need to find z α/2, the critical Z-value ●z α/2 is the value such that P(Z ≥ z α/2 ) = α/2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 30 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 to

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 31 of 39 Chapter 9 – Section 1 ●Learning objectives  Compute a point estimate of the population mean  Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known)  Understand the role of margin of error in constructing a confidence interval  Determine the sample size necessary for estimating the population mean within a specified margin of error

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 32 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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 33 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) ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 34 of 39 Chapter 9 – Section 1 ●The margin of errors would be  σ / √ n for 90% confidence intervals  σ / √ n for 95% confidence intervals  σ / √ n for 99% confidence intervals

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 35 of 39 Chapter 9 – Section 1 ●Learning objectives  Compute a point estimate of the population mean  Construct and interpret a confidence interval about the population mean (assuming the population standard deviation is known)  Understand the role of margin of error in constructing a confidence interval  Determine the sample size necessary for estimating the population mean within a specified margin of error

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 36 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 – 2 nd Edition – Chapter 9 Section 1 – Slide 37 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 ●Solving for n would get us n = (1.96 6) 2 or that n = 138 cars would be needed

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 9 Section 1 – Slide 38 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 ●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 – 2 nd Edition – Chapter 9 Section 1 – Slide 39 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