8.3: Confidence Intervals for Means 2.15.2018

Means vs. Proportions We started with calculating confidence intervals for proportions Usually categorical/discrete variables E.g. proportion of cards that are red, proportion of the time that you roll a one/six, proportion of people with blue eyes, etc. Now we’re going to do confidence intervals for means Usually used for continuous variables Could also be used for discrete as well (but not categorical)

Two Very Different Situations We know the population standard deviation We DON’T know the population standard deviation This is basically what we have done in previous sections We use the population standard deviation to calculate the standard deviation of the sampling distribution Critical values follow the normal distribution (just like they did for proportions in 8.2) We have to estimate the standard deviation from the sample We use the sample standard deviation to estimate the standard deviation of the distribution Critical values follow the t distribution (NEW)

Known Population Standard Deviation This is the easier case CI=(point estimate) ± (critical value)(St. dev of distribution) Standard deviation of the distribution equals standard deviation of the population divided by the square root of n

Example Researchers would like to estimate the mean cholesterol level of a particular variety of monkey that is often used in laboratory experiments. They take a sample of 200 monkeys, and find a mean of 34.2 mg/dl. A previous study has found that the standard deviation of cholesterol levels is about 5 mg/dl. Find the 95% confidence interval

Example (34.2) ± (critical value)(5 / 200 ) (34.2) ± (2)(.354) (34.2) ± (.707) 33.493—34.907

Back to the Monkeys Researchers would like to estimate the mean cholesterol level of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1 mg/dl of the true value at a 95% confidence level. A previous study has found that the standard deviation of cholesterol levels is about 5 mg/dl. What sample size do they need to take?

Back to the Monkeys ME=(critical value)(st dev of distribution) 1=(2)(5 / 𝑛 ) .5=(5 / 𝑛 ) 𝑛 =5/.5 𝑛 =10 N=100

Back to the Monkeys (again) What if we wanted a 90% confidence level instead of a 95% confidence level

Back to the Monkeys (again) What if we wanted a 90% confidence level instead of a 95% confidence level 1=(critical value)(5 / 𝑛 ) Critical value = invnorm(.05,0,1) = 1.645 1=(1.645)(5 / 𝑛 ) .608=(5 / 𝑛 ) 𝑛 = 8.224 N=67.63 So they should sample at least 68 monkeys

When we DON’T know the population standard deviation This is the harder situation And, unfortunately, more common Note: this only applies to means, NOT PROPORTIONS The difference occurs in the way that we find critical values Before, when we knew the standard deviation of the population, we could easily calculate how many standard deviations we needed to go out from the mean to capture the desired proportion of the data .95 for a 95% confidence interval gave us a critical value of 2 (1.96)

Critical Values So now we are replacing σ (pop. Standard deviation) with 𝑆 𝑥 (sample standard deviation) Z was our critical value For proportions For means when we know σ But now our critical value is t

Why do we need a new critical value? If we take a sample from a population, and measure the standard deviation of the sample, it is (on average) going to vary more than the standard deviation of the population. See the simulation below (

Critical Values Z was our critical value For proportions For means when we know σ But now our critical value is t Why didn’t we need to do this for proportions?

Critical Values Z was our critical value For proportions For means when we know σ But now our critical value is t Why didn’t we need to do this for proportions? Because 𝑝 ≈𝑝, but 𝑠 𝑥 ≠𝜎

The t distribution This new distribution is called the t distribution It looks pretty similar to a normal distribution The difference is that it has more area in the tails than a normal distribution Particularly with smaller sample sizes

So…how do we get critical values? Instead of using standard normal probabilities Table A or invnorm() Now we will use the t distribution T distribution only has one input: ‘degrees of freedom’ or ‘df’ Sample size minus 1

What are degrees of freedom? This is a common question that people want to ask—what do the degrees of freedom mean? The teacher’s edition, unsatisfyingly, says: “Unfortunately, there is no simple answer. For now, simply explain that the shape and spread of the t distributions depend on the degrees of freedom, which depend on the sample size. The larger the sample size—and the more degrees of freedom—the closer the t-distributions come to the standard Normal distribution”

What are degrees of freedom? That answer doesn’t help much Unfortunately, the answer has to do with ideas that are above the pay grade of AP statistics One fairly accurate way of thinking about it is that it is the number of pieces of information that we have that allow us to make an estimate For more precise (and more complicated) explanations: https://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics) In practice, you don’t need to know what degrees of freedom really mean—you just need to know how many there are

So…how do we get critical values? Instead of using standard normal probabilities Now we will use the t distribution T distribution only has one input: df 2 options: Use Table B (if you have a TI-84 or above): invT()

Using Table B Table B is (I think) more intuitive to use than Table A, so it is a viable option Find your degrees of freedom (df) on the rows, and your confidence level C on the collumns Where they intersect tells you the critical value

Using Table B

Using Table B If you look down at the bottom, the last row has an infinity sign As df is infinitely large, then the t distribution becomes a normal distribution So you could actually use table B for proportions also, by looking at that row For example, for a confidence level of 95%, it says that the critical value would be 1.96 So if you’re going to use a table instead of your calculator, you might just want to use Table B, rather than Table A and Table B

Using InvT Only if you have a TI-84 or above Sorry TI-83 owners Works the same way as Invnorm You plug in the area (to the left) And then the degrees of freedom Table B told us the critical value was 2.201 InvT tells us that the critical value is 2.200985143 Rounds to 2.201

Reminder about normality We can only use these procedures if the sample size is big enough If n<15, only if the data appear to be approximately Normal (symmetric, single-peaked, no outliers) If 15≤n<30, can use unless there are significant outliers or skewness If n≥30, go for it! Sidenote: if you are choosing the sample size for a study—it should probably be bigger than 30 so that this isn’t a problem

An Example John Isner is a professional tennis player. I take an SRS of size 51 of his first serves and measure their speed in miles/hour. In the sample, the mean is 124 mph and the standard deviation is 8 mph Find the 98% confidence interval for the mean

One-Sample t Interval for the population mean (Point Estimate) ± ME (Point estimate) ± (critical value)(St. dev)

One Sample t interval for the population mean (Point Estimate) ± ME (Point estimate) ± (critical value)(St. dev) 124 ± (2.403)( 𝑆 𝑥 𝑛 ) 124 ± (2.403)( 8 51 ) 124 ± (2.403)(1.12) 124 ± 2.691 121.31—126.69 We are 98% confident that John Isner’s mean first serve speed is between 121.31 and 126.69 mph

Different Example A random sample of 30 students received SAT math scores with a mean of 580 and a standard deviation of 80 Find an 80% confidence interval for the true mean of SAT math scores

One sample t interval for the population mean (Point Estimate) ± ME (Point estimate) ± (critical value)(St. dev) 580 ± (1.311)( 𝑆 𝑥 𝑛 ) 580 ± (1.311)( 80 30 ) 580 ± (1.311)(14.606) 580 ± 19.148 560.85—599.15 We are 80% confident that the mean SAT math score is between 560.85 and 599.15

Using Your Calculator Your calculator can do this too Just like it could for proportions STAT---TESTS--- “Tinterval” It will give you the choice of “data” or “stats” Data means that it will use the data that you have entered in L1 Use this if you have the raw data Stats means that you will enter 𝑥 and 𝑆 𝑥 Use this if you know these values And actually, even if you do enter data into L1, and then do 1-Var Stats, it will auto-populate into Tinterval

Using Your Calculator Again, you need to know how to do it by hand In case you’re asked to specifically state the critical value, or point estimate, etc. If you do it on your calculator, to get full credit on the test: clearly state how you know that the sampling distribution is normal (checking for normality) And state what type of interval you’re creating 1-sample t interval for population mean 1-proportion z interval, etc. Or write the equation out as if you were doing it by hand

An Example I take a random sample (with replacement) of 15 of your raw chapter 7 test scores. None of the scores are outliers, and the distribution of the sample data is roughly symmetric. The mean of the 15 scores is 57.8, with a standard deviation of 18.09 Calculate a 90% confidence interval for the true mean of the chapter 7 test scores

An Example I take a random sample (with replacement) of 15 of your raw chapter 7 test scores. Assume that none of the scores are outliers, and the distribution of the sample data is roughly symmetric. The mean of the 15 scores is 57.8, with a standard deviation of 18.09 Calculate a 90% confidence interval for the true mean of the chapter 7 test scores (49.574 , 66.026) P.S. The true mean was 59.9

Last Problem! 319 328 321 324 322 320 324 321 320 324 322 317 321 320 322 318 326 316 316 326 325 320 316 319 319 321 322 319 326 320 324 320 318 321 322 318 The data to the left represent a sample of aspirin tablets. These measurements are how much acetylsalicylic acid (in mg) is found in each tablet Construct a 99% confidence interval for the mean amount of acid found in aspirin tablets

Last Problem! 319 328 321 324 322 320 324 321 320 324 322 317 321 320 322 318 326 316 316 326 325 320 316 319 319 321 322 319 326 320 324 320 318 321 322 318 The data to the left represent a sample of aspirin tablets. These measurements are how much acetylsalicylic acid (in mg) is found in each tablet Construct a 99% confidence interval for the mean amount of acid found in aspirin tablets (319.65 , 322.4) True value (according to the manufacturer) is 320