Chapter 10Chapter 10 confidence intervals confidence intervals

For MeansFor MeansFor proportionsFor proportions

ActivityActivity

Roll your real die 50 times and record Roll your real die 50 times and record each number.each number.

Find the mean of the die.Find the mean of the die. Find the standard deviation of the die.Find the standard deviation of the die. You know that this die averages 3.5. Is You know that this die averages 3.5. Is

there evidence that this is not true?there evidence that this is not true?

Is the mean 3.5?Is the mean 3.5?

Construct a 95% confidence interval for Construct a 95% confidence interval for the true mean of the die.the true mean of the die. What would the requirements/assumptions What would the requirements/assumptions

be for this interval? (HINT--SIN!)be for this interval? (HINT--SIN!)

95% Confidence Interval.95% Confidence Interval.

Construct a 95% confidence interval Construct a 95% confidence interval using Z*using Z*

How many of these intervals captured U, How many of these intervals captured U, which we know to be 3.5which we know to be 3.5

Construct a 95% C.I. using T*. How Construct a 95% C.I. using T*. How many of these intervals captured U?many of these intervals captured U?

What is the meaning of these intervals?

Meaning of a 95% C.I.Meaning of a 95% C.I. The meaning is NOT:The meaning is NOT:

95% of all rolls are between 3 and 495% of all rolls are between 3 and 4

It is:It is: I am 95% confident that my interval captured I am 95% confident that my interval captured

the mean because if this process were to be the mean because if this process were to be done repeatedly, about 95% of all intervals done repeatedly, about 95% of all intervals would capture the true mean of the die.would capture the true mean of the die.

90% Confidence Interval.90% Confidence Interval.

Now construct a 90% confidence interval Now construct a 90% confidence interval for the same data using Z*. See any for the same data using Z*. See any differences?differences?

Now a 90% CI using T*. What do you see?Now a 90% CI using T*. What do you see?

99% C.I.99% C.I. Construct a 99% C.I. using Z*. Compare Construct a 99% C.I. using Z*. Compare

with the 90% and 95% C.I.s. What do you with the 90% and 95% C.I.s. What do you notice?notice?

Construct a 99% C.I. using T*. Compare Construct a 99% C.I. using T*. Compare with the 90% and 95% C.I.s. What do you with the 90% and 95% C.I.s. What do you notice?notice?

What are some ways to What are some ways to shrink your interval?shrink your interval?

Lower confidence.Lower confidence. Higher sample size.Higher sample size.

Confidence intervals—Confidence intervals—Day 2Day 2

Take your die—the one you made--and roll it Take your die—the one you made--and roll it 25 times.25 times.

What is the mean?What is the mean? What is the standard deviation.What is the standard deviation. Make a 95% Z-interval.Make a 95% Z-interval. Make a 95% T-interval.Make a 95% T-interval. Did you meet the requirements? What about Did you meet the requirements? What about

the normal part, how do you judge that.the normal part, how do you judge that.

Is it OK to use a T procedure here?

How about here? Why?

Is a t procedure Ok here?

Under What conditions would this distribution be OK?

Is this normal plot acceptable?

How about this plot?

Is your die fair?Is your die fair? What does your interval say about your die?What does your interval say about your die?

Do you think that it is fair?Do you think that it is fair? Could it average 3.5 but you just got a weird Could it average 3.5 but you just got a weird

sample?sample?

You should know:You should know: What your confidence interval means.What your confidence interval means. What the margin of error is.What the margin of error is. How to calculate sample size requirements.How to calculate sample size requirements.

How do we find the exact How do we find the exact sample size we want?sample size we want?

Z*(Z*(σσ/√n) = margin of error/√n) = margin of error

OROR

T*(s/√n) = margin of errorT*(s/√n) = margin of error

Type 1 and type 2 errorsType 1 and type 2 errors

Examine your die Examine your die data. Do you think data. Do you think that your conclusion that your conclusion about your die is about your die is right or wrong? Could right or wrong? Could you have made an you have made an error?error?

What are the What are the chances of that chances of that error?error?

Die is fairDie is fair—you think —you think it is too—it is too—good!good!

Die is fairDie is fair—you think —you think it’s not—it’s not—type 1 type 1 error.error.

Die is Die is unfair—unfair—you think you think it’s O.K.—it’s O.K.—type 2 type 2 errorerror

Die is Die is unfair—unfair—you you detected detected that—that—good!good!

Confidence intervals—Confidence intervals—Day 3Day 3

Roll your die 60 times to see the proportion Roll your die 60 times to see the proportion of 5s that you get.of 5s that you get.

Write down the number of 5s that you get. Write down the number of 5s that you get. Did you get an unusual amount? Unusually Did you get an unusual amount? Unusually high or low?high or low?

Make a confidence interval for the Make a confidence interval for the proportion of 5s your die would get if you proportion of 5s your die would get if you rolled it indefinitely.rolled it indefinitely.

What are the requirements for this What are the requirements for this situation?situation?

Confidence intervalsConfidence intervalsproportionsproportions

S representative SampleS representative Sample N np>10 and n(1-p)> 10N np>10 and n(1-p)> 10 A andA and P opulation 10X bigger than P opulation 10X bigger than

sample size.sample size.

CI for a proportion—there are no Ts for this.

CI proportionsCI proportions

Make a 90%, 95% and 99% confidence Make a 90%, 95% and 99% confidence interval.interval.

Is your die fair based on this criteria.Is your die fair based on this criteria. How big a sample do you need to reduce How big a sample do you need to reduce

the margin of error to less than 3%?the margin of error to less than 3%? Did you potentially make an error? Type Did you potentially make an error? Type

1? Type 2?1? Type 2? What do these intervals mean?What do these intervals mean?