Chapter 10 confidence intervals  For Means  For proportions

Activity  Roll your real die 50 times and record each number.  Find the mean of the die.  Find the standard deviation of the die.  You know that this die averages 3.5. Is there evidence that this is not true?

Is the mean 3.5?  Construct a 95% confidence interval for the true mean of the die.  What would the requirements/assumptions be for this interval? (HINT--SIN!)

95% Confidence Interval.  Construct a 95% confidence interval using Z*  How many of these intervals captured U, which we know to be 3.5  Construct a 95% C.I. using T*. How many of these intervals captured U?

What is the meaning of these intervals?

Meaning of a 95% C.I.  The meaning is NOT:  95% of all rolls are between 3 and 4  It is:  I am 95% confident that my interval captured the mean because if this process were to be done repeatedly, about 95% of all intervals would capture the true mean of the die.

90% Confidence Interval.  Now construct a 90% confidence interval for the same data using Z*. See any differences?  Now a 90% CI using T*. What do you see?

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

What are some ways to shrink your interval?  Lower confidence.  Higher sample size.

Confidence intervals— Day 2  Take your die—the one you made--and roll it 25 times.  What is the mean?  What is the standard deviation.  Make a 95% Z-interval.  Make a 95% T-interval.  Did you meet the requirements? What about 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?  What does your interval say about your die?  Do you think that it is fair?  Could it average 3.5 but you just got a weird sample?  You should know:  What your confidence interval means.  What the margin of error is.  How to calculate sample size requirements.

How do we find the exact sample size we want? Z*(σ/√n) = margin of error OR T*(s/√n) = margin of error

Type 1 and type 2 errors  Examine your die data. Do you think that your conclusion about your die is right or wrong? Could you have made an error?  What are the chances of that error? Die is fair—you think it is too—good! Die is fair—you think it’s not—type 1 error. Die is unfair— you think it’s O.K.— type 2 error Die is unfair— you detected that— good!

Confidence intervals— Day 3  Roll your die 60 times to see the proportion of 5s that you get.  Write down the number of 5s that you get. Did you get an unusual amount? Unusually high or low?  Make a confidence interval for the proportion of 5s your die would get if you rolled it indefinitely.  What are the requirements for this situation?

Confidence intervals proportions  S representative Sample  N np>10 and n(1-p)> 10  A and  P opulation 10X bigger than sample size.

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

CI proportions  Make a 90%, 95% and 99% confidence interval.  Is your die fair based on this criteria.  How big a sample do you need to reduce the margin of error to less than 3%?  Did you potentially make an error? Type 1? Type 2?  What do these intervals mean?