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?