1. Chapter 9 Confidence Intervals

2. Suppose we wanted to estimate the proportion of blue candies in a VERY large bowl. How might we go about estimating this proportion? We could take a sample of candies and compute the proportion of blue candies in our sample. We would have a sample proportion or a statistic – a single value for the estimate.

3. Point Estimate A single number (a statistic) based on sample data that is used to estimate a population characteristic But not always to the population characteristic due to sampling variation “point” refers to the single value on a number line. Different samples may produce different statistics. Population characteristic

4. The paper “U.S. College Students’ Internet Use: Race, Gender and Digital Divides” (Journal of Computer-Mediated Communication, 2009) reports the results of 7421 students at 40 colleges and universities. (The sample was selected in such a way that it is representative of the population of college students.) The authors want to estimate the proportion (p) of college students who spend more than 3 hours a day on the Internet. 2998 out of 7421 students reported using the Internet more than 3 hours a day. This is a point estimate for the population proportion of college students who spend more than 3 hours a day on the Internet. p = 2998/7421 =.404

5. The paper “The Impact of Internet and Television Use on the Reading Habits and Practices of College Students” (Journal of Adolescence and Adult Literacy, 2009) investigates the reading habits of college students. The following observations represent the number of hours spent on academic reading in 1 week by 20 college students. 1.73.84.79.611.712.3 12.412.613.4 14.114.215.815.918.719.421.221.923.328.2 The dotplot suggest this data is approximately symmetrical. If a point estimate of , the mean academic reading time per week for all college students, is desired, an obvious choice of a statistic for estimating  is the sample mean x ̅. However, there are other possibilities – a trimmed mean or the sample median.

6. College Reading Continued... 1.73.84.79.611.712.3 12.412.613.4 14.114.215.815.918.719.421.221.923.328.2 The mean of the middle 16 observations. So which of these point estimates should we use?

7. Choosing a Statistic for Computing an Estimate Choose a statistic that is unbiased (accurate) A statistic whose mean value is equal to the value of the population characteristic being estimated is said to be an unbiased statistic. Unbiased, since the distribution is centered at the true value Biased, since the distribution is NOT centered at the true value

8. Choosing a Statistic for Computing an Estimate Choose a statistic that is unbiased (accurate) Choose a statistic with the smallest standard deviation Unbiased, but has a larger standard deviation so it is not as precise. Unbiased, but has a smaller standard deviation so it is more precise. If the population distribution is normal, then x ̅ has a smaller standard deviation than any other unbiased statistic for estimating .

9. 1.When the population distribution is normal, then x ̅ has a smaller standard deviation than any other unbiased statistic. 2.When the population distribution is symmetric with heavy tails compared to the normal curve, a trimmed mean is a better statistic than x ̅ for estimating . 3.To estimate the population variance,  2, use the sample variance:

10. Suppose we wanted to estimate the proportion of blue candies in a VERY large bowl. We could take a sample of candies and compute the proportion of blue candies in our sample. How much confidence do you have in the point estimate? Would you have more confidence if your answer were an interval?

11. Confidence intervals A confidence interval (CI) for a population characteristic is an interval of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence, the actual value of the characteristic will be between the lower and upper endpoints of the interval. The primary goal of a confidence interval is to estimate an unknown population characteristic.

12. Rate your confidence 0 – 100% How confident (%) are you that you can... Guess my age within 10 years?... within 5 years?... within 1 year? What does it mean to be within 10 years? What happened to your level of confidence as the interval became smaller?

13. Confidence level The confidence level associated with a confidence interval estimate is the success rate of the method used to construct the interval. If this method was used to generate an interval estimate over and over again from different samples, in the long run 95% of the resulting intervals would include the actual value of the characteristic being estimated. Our confidence is in the method – NOT in any one particular interval! The most common confidence levels are 90%, 95%, and 99% confidence.

14. Recall the General Properties for Sampling Distributions of p 1. 2. As long as the sample size is less than 10% of the population 3. As long as n is large (np > 10 and n (1-p) > 10) the sampling distribution of p is approximately normal. These are the conditions that must be true in order to calculate a large-sample confidence interval for p

15. Let’s develop the equation for the large-sample confidence interval. To begin, we will use a 95% confidence level. Use the table of standard normal curve areas to determine the value of z* such that a central area of.95 falls between –z* and z*. -1.961.96 Central Area =.95 Upper tail area =.025Lower tail area =.025 95% of these values are within 1.96 of the mean. We can generalize this to normal distributions other than the standard normal distribution – About 95% of the values are within 1.96 standard deviations of the mean 0 For large random samples, the sampling distribution of p is approximately normal. So about 95% of the possible p will fall within

16. Developing a Confidence Interval Continued... If p is within of p, this means the interval will capture p. And this will happen for 95% of all possible samples!

17. Developing a Confidence Interval Continued... Here is the mean of the sampling distribution p Approximate sampling distribution of p This line represents 1.96 standard deviations below the mean. This line represents 1.96 standard deviations above the mean. Suppose we get this p p Create an interval around p Notice that the length of each half of the interval equals This p fell within 1.96 standard deviations of the mean AND its confidence interval “captures” p. Suppose we get this p and create an interval p This p fell within 1.96 standard deviations of the mean AND its confidence interval “captures” p. Suppose we get this p and create an interval p This p doesn’t fall within 1.96 standard deviations of the mean AND its confidence interval does NOT “capture” p. Using this method of calculation, the confidence interval will not capture p 5% of the time. When n is large, a 95% confidence interval for p is

18. The diagram to the right is 100 confidence intervals for p computed from 100 different random samples. Note that the ones with asterisks do not capture p. If we were to compute 100 more confidence intervals for p from 100 different random samples, would we get the same results?

19. The general formula for a confidence interval for a population proportion p when if the sample is selected without replacement, the sample size is small relative to the population size (at most 10% of the population) The Large-Sample Confidence Interval for p Now let’s look at a more general formula. p is the sample proportion from a random sample the sample size n is large (np > 10 and n(1-p) > 10), and

20. The Large-Sample Confidence Interval for p The general formula for a confidence interval for a population proportion p... is point estimate Estimate of the standard deviation of p or standard error The standard error of a statistic is the estimated standard deviation of the statistic.

21. The Large-Sample Confidence Interval for p The general formula for a confidence interval for a population proportion p... is This is called the bound on the error estimation. The 95% confidence interval is based on the fact that, for approximately 95% of all random samples, p is within the bound on error estimation of p. This is also called the margin of error.

22. The article “How Well Are U.S. Colleges Run?” (USA Today, February 17, 2010) describes a survey of 1031 adult Americans. The survey was carried out by the National Center for Public Policy and the sample was selected in a way that makes it reasonable to regard the sample as representative of adult Americans. Of those surveyed, 567 indicated that they believe a college education is essential for success. What is a 95% confidence interval for the population proportion of adult Americans who believe that a college education is essential for success? Before computing the confidence interval, we need to verify the conditions. The point estimate is

23. College Education Continued... What is a 95% confidence interval for the population proportion of adult Americans who believe that a college education is essential for success? Conditions : 2) The sample size of n = 1031 is much smaller than 10% of the population size (adult Americans). 3) The sample was selected in a way designed to produce a representative sample. So we can regard the sample as a random sample from the population. 1) np = 1031(.55) = 567 and n(1-p) = 1031(.45) = 364, since both of these are greater than 10, the sample size is large enough to proceed. All our conditions are verified so it is safe to proceed with the calculation of the confidence interval.

24. College Education Continued... What is a 95% confidence interval for the population proportion of adult Americans who believe that a college education is essential for success? Calculation : Conclusion: We are 95% confident that the population proportion of adult Americans who believe that a college education is essential for success is between 52.1% and 57.9% What does this interval mean in the context of this problem?

25. College Education Revisited... A 95% confidence interval for the population proportion of adult Americans who believe that a college education is essential for success is: Compute a 90% confidence interval for this proportion. Compute a 99% confidence interval for this proportion. What do you notice about the relationship between the confidence level of an interval and the width of the interval? Recall the “Rate your Confidence” Activity

26. Choosing a Sample Size The bound on error estimation for a 95% confidence interval is If we solve this for n... Before collecting any data, an investigator may wish to determine a sample size needed to achieve a certain bound on error estimation. What value should be used for the unknown value p? Sometimes, it is feasible to perform a preliminary study to estimate the value for p. In other cases, prior knowledge may suggest a reasonable estimate for p. If there is no prior knowledge and a preliminary study is not feasible, then the conservative estimate for p is 0.5.

27. Why is the conservative estimate for p = 0.5?.1(.9) =.09.2(.8) =.16.3(.7) =.21.4(.6) =.24.5(.5) =.25 By using.5 for p, we are using the largest value for p(1 – p) in our calculations.

28. In spite of the potential safety hazards, some people would like to have an internet connection in their car. Determine the sample size required to estimate the proportion of adult Americans who would like an internet connection in their car to within 0.03 with 95% confidence. What value should be used for p? This is the value for the bound on error estimate B. Always round the sample size up to the next whole number.

29. Now let’s look at confidence intervals to estimate the mean  of a population.

30. Confidence intervals for  when  is known The general formula for a confidence interval for a population mean  when... 1) x is the sample mean from a random sample, 2) the sample size n is large (n > 30), and 3) , the population standard deviation, is known is These are the properties of the sampling distribution of x. Is this typically known? This confidence interval is appropriate even when n is small, as long as it is reasonable to think that the population distribution is normal in shape. Point estimate Standard deviation of the statistic Bound on error of estimation

31. Cosmic radiation levels rise with increasing altitude, promoting researchers to consider how pilots and flight crews might be affected by increased exposure to cosmic radiation. A study reported a mean annual cosmic radiation dose of 219 mrems for a sample of flight personnel of Xinjiang Airlines. Suppose this mean is based on a random sample of 100 flight crew members. Let  = 35 mrems. Calculate and interpret a 95% confidence interval for the actual mean annual cosmic radiation exposure for Xinjiang flight crew members. 1)Data is from a random sample of crew members 2)Sample size n is large (n > 30) 3)  is known First, verify that the conditions are met.

32. Cosmic Radiation Continued... Let x = 219 mrems n = 100 flight crew members  = 35 mrems. Calculate and interpret a 95% confidence interval for the actual mean annual cosmic radiation exposure for Xinjiang flight crew members. What does this mean in context? We are 95% confident that the actual mean annual cosmic radiation exposure for Xinjiang flight crew members is between 212.14 mrems and 225.86 mrems. What would happen to the width of this interval if the confidence level was 90% instead of 95%?

33. Confidence intervals for  when  is unknown When  is unknown, we use the sample standard deviation s to estimate . In place of z-scores, we must use the following to standardize the values: The use of the value of s introduces extra variability. Therefore the distribution of t values has more variability than a standard normal curve.

34. Important Properties of t Distributions 1)The t distribution corresponding to any particular number of degrees of freedom is bell shaped and centered at zero (just like the standard normal (z) distribution). 2)Each t distribution is more spread out than the standard normal distribution. z curve t curve for 2 df Why is the z curve taller than the t curve for 2 df? 0 t distributions are described by degrees of freedom (df).

35. Important Properties of t Distributions Continued... 3) As the number of degrees of freedom increases, the spread of the corresponding t distribution decreases. t curve for 8 df t curve for 2 df 0

36. Important Properties of t Distributions Continued... 3) As the number of degrees of freedom increases, the spread of the corresponding t distribution decreases. 4) As the number of degrees of freedom increases, the corresponding sequence of t distributions approaches the standard normal distribution. z curve t curve for 2 df t curve for 5 df For what df would the t distribution be approximately the same as a standard normal distribution? 0

37. Confidence intervals for  when  is unknown The general formula for a confidence interval for a population mean  based on a sample of size n when... 1) x is the sample mean from a random sample, 2) the population distribution is normal, or the sample size n is large (n > 30), and 3) , the population standard deviation, is unknown is Where the t critical value is based on df = n - 1. This confidence interval is appropriate for small n ONLY when the population distribution is (at least approximately) normal. t critical values are found in Table 3

38. The article “Chimps Aren’t Charitable” (Newsday, November 2, 2005) summarized the results of a research study published in the journal Nature. In this study, chimpanzees learned to use an apparatus that dispersed food when either of two ropes was pulled. When one of the ropes was pulled, only the chimp controlling the apparatus received food. When the other rope was pulled, food was dispensed both to the chimp controlling the apparatus and also a chimp in the adjoining cage. The accompanying data represent the number of times out of 36 trials that each of seven chimps chose the option that would provide food to both chimps (charitable response). 23222124192020 Compute a 99% confidence interval for the mean number of charitable responses for the population of all chimps. First verify that the conditions for a t-interval are met.

39. The plot is reasonable straight, so it seems plausible that the population distribution of number of charitable responses is approximately normal. Number of Charitable Responses Normal Scores Chimps Continued... 23222124192020 Since n is small, we need to verify if it is plausible that this sample is from a population that is approximately normal. Let’s use a normal probability plot. Let’s suppose it is reasonable to regard this sample of seven chimps as representative of the chimp population.

40. Chimps Continued... 23222124192020 x = 21.29 and s = 1.80df = 7 – 1 = 6 We are 99% confident that the mean number of charitable responses for the population of all chimps is between 18.77 and 23.81.

41. The bound on error of estimation associated with a 95% confidence interval is Solve this for n: Choosing a Sample Size We can use this to find the necessary sample size for a particular bound on error of estimation. This requires  to be known – which is rarely the case! When  is unknown, a preliminary study can be performed to estimate  OR make an educated guess of the value of . A rough estimate for  (used with distributions that are not too skewed) is the range divided by 4.

42. The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true population mean. How large a sample should be used to be 95% confident of achieving this level of accuracy? The financial aid office is believes that the amount spent on books varies with most values between $150 to $550. To estimate  :

43. The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true population mean. How large a sample should be used to be 95% confident of achieving this level of accuracy? Always round sample size up to the next whole number!