1. 8.3: Confidence Intervals for Means

2. 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)

3. 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)

4. 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

5. 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

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

7. 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?

8. 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

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

10. 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

11. 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)

12. 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

13. 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 (

14. 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?

15. 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

16. 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

17. 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

18. 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”

19. 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:
In practice, you don’t need to know what degrees of freedom really mean—you just need to know how many there are

20. 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()

21. 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

22. Using Table B

23. 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

24. 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
Rounds to 2.201

25. 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

26. 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

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

28. 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)( )
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 and mph

29. 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

30. 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)( )
580 ± (1.311)(14.606)
580 ±
560.85—599.15
We are 80% confident that the mean SAT math score is between and

31. 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

32. 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

33. 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

34. 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
( , )
P.S. The true mean was 59.9

35. Last Problem!
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

36. Last Problem!
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
( , 322.4)
True value (according to the manufacturer) is 320