1. Confidence Intervals (cont.)

Transcription

1 Math 1125-Introductory Statistics Lecture 23 11/1/06 1. Confidence Intervals (cont.) Let s review. We re in a situation, where we don t know µ, but we have a number from a normal population, either an x or an x. We would expect that these numbers are closer to µ than they are far, but we ll never know for sure. We re going to assume that the possible µ s for the populations that our x or x came from must be normally distributed with standard deviation σ. The symmetry in z-score formula suggests that we can do this, but if you think about it too deeply, logical concerns will arise. A confidence interval is a way of approximating the value of a population mean µ from a sample mean x of size n. Essentially, we re going to take the middle 90%, or some other similar number, and say which possible µ s lie in that middle 90%. If we know what the population standard deviation σ, then the confidence interval can be found as follows. We compute the standard deviation for the sample means (1) σ x = σ n, and then the confidence interval is given by (2) x z σ x µ x + z σ x In this formula, z is the worst z-score associated with the middle X% for X% confidence. The z-scores we ll use are found at the bottom of the t-table that we ll discuss in a bit. We can, if we want, find the standard deviation for the sample means and the confidence interval at the same time, and use the formula (3) x z σ µ x + z σ. n n For example, suppose we wanted to find the average distance between the pupils of adult females in the United States. Let s say we took a sample of size n = 70, and found a sample mean of x = 2.93 inches. We ll assume that we know the population standard deviation of σ = 0.35 inches. Find the 95% confidence interval. One thing we ll need is (4) σ x = =

2 2 Next we ll need the worst z-score at the edges of the middle 95%. This corresponds to an area of in our z-table, or at the top of the new t-table, you ll see a confidence level of 95%, and at the very bottom of this column, you ll see a z-score of z =1.96. We have everything we need. (5) z or (6) 2.85 µ 3.01 This is the 95% confidence interval. 2. The t-distribution In the computations done so far, there was one unrealistic aspect to all of it. We needed to know the true population standard deviation. It seems as though, if we didn t know µ, we probably wouldn t know σ either. There are situations, where we would know only σ, but most of the time we won t. What should we do? Since we don t have σ, the next best thing is the sample standard deviation s. We can actually find this from the sample. In fact, we did this earlier in the semester. The problem is that s is probably not equal to σ. If the population is approximately normal, however, it s possible to compute a probability distribution on s. From there, we can figure out where the middle X% is. Well, we can t, but lot s of people can. W. T. Gosset, who worked for the Guiness Brewery, was the first to work out the probabilities. He found that if we use s instead of σ, then we need to use a number, which we ll call a t-score, that s a bit bigger than the z-score we would use normally. Our new formula looks like (7) x t s s µ x + t n n The t we use is the worst t-score associated with the middle X%. This is found in the t-table as follows. The probabilities depend on the sample size (since smaller samples mean more variability in s). You may remember seeing n 1 in our computations of s 2. We ll see this again, and call it the degrees of freedom. That is, the degrees of freedom are (8) df = n 1 We find the worst t score in the table according to the level of confidence and the degrees of freedom.

3 3 3. Some examples Suppose we re doing research on unicorn horns, and have found ten of them. It s reasonable to assume that the lengths of unicorn horns are normally distributed, but otherwise, we only have specific information about these ten horns. We find that the mean length is x =8.5 inches with a standard deviation of s = 1.2 inches. Let s construct a 90% confidence interval. The level of confidence is 90% and the degrees of freedom are df =10 1 = 9. Looking in the table, we find t =1.83. This is the worst t-score associated with the middle 90%. The 90% confidence interval is (9) or µ (10) 7.81 µ Quiz 23 Suppose that I ve designed a drug Howisinol that is supposed to lower blood pressure. I believe it s reasonable to assume a normal distribution. I want to know how well it works. I take a sample of size n = 20, and find that on this group of 20 people, their blood pressure goes down by an average x =22.5points. The standard deviation for the sample is s =4.3. Find a 90% confidence interval for the population mean. 1. What is df? 2. What is the worst t-score? The confidence interval will take the form A µ B. 3. Find A. 4. Find B. (Homework 23 starts on next page)

4 4 5. Homework 23 Round all of your answers to two decimal places. Problems 1-3, refer to the drug Howisinol described in Quiz 23, and you re going to find the 99% confidence interval, A µ B. 1. What s the worst t-score? 2. What s A? 3. What s B? For problems 4-12, we have a test whose scores we ll assume are approximately normally distributed. We try it out on 20 people, and their average score is x =57.7 with a standard deviation of s = We don t know exactly what the population mean will be when we give the test to everyone in the population. Based on the information we have, what is the most reasonable single value for µ? 5. What s df? For problems 6-8, you re going to find the 90% confidence interval, A µ B. 6. What s the worst t-score? 7. What s A? 8. What s B? For problems 9-11, you re going to find the 99% confidence interval, A µ B. 9. What s the worst t-score? 10. What s A? 11. What s B? (cont. on next page)

5 5 For problems 12-20, we can reasonably assume that distance between female jungle dogs eyes are normally distributed. We are going to make sunglasses for them, but we can only afford to make one size, and we want them to fit as many dogs as possible. We re going to start our research with a confidence interval. We sample 15 dogs and find an average distance of x =1.2 inches and standard deviation of s =0.3 inches. 12. Based only on this information, what distance should we base our sunglass size on? 13. What s df? For problems 14-16, you re going to find the 95% confidence interval, A µ B. 14. What s the worst t-score? 15. What s A? 16. What s B? For problems 17-19, you re going to find the 98% confidence interval, A µ B. 17. What s the worst t-score? 18. What s A? 19. What s B? 20. In the study of some population, we take a sample, find the mean and standard deviation of the sample, and construct a confidence interval. If our confidence intervals comes out to be 12.0 µ 14.0, what must the sample mean, x, have been? Bye.