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8.1 Estimating When is Known (Page 1 of 25) 8.1 Estimating When is Known Assumptions about the random variable x 1. We have a simple random sample of size n drawn from the population of x values. 2. The value of , the population standard deviation is known. 3. If the x distribution is normal, then our methods work for any sample size n. 4. If x has an unknown distribution, then we require the sample size n 30 . However, if the x distribution is not moundshaped, then a sample size of 50 or 100 may be needed. Point Estimate An estimate of a population given by a single number is called a point estimate of that population parameter. For Example: x is a point estimate for . s is a point estimate for . Margin of Error The margin of error in using x as a point estimate for is given by E x . A point estimate is not very useful unless we have some kind of measure of how “good” it is. This “measure of goodness” is expressed as a confidence interval. 8.1 Estimating When is Known (Page 2 of 25) Confidence Interval and Level of Confidence Suppose 100 students at Palomar were randomly chosen and their heights were measured yielding a [sample] mean of 5.72 ft with a margin of error of 0.08 ft. Consider the following statements: 1. The population mean is approximately 5.72 feet. 2. There is a 95% probability that the population mean is between 5.64 ft and 5.80 ft. P(5.64 ft 5.80 ft) 0.95 3. At a 95% level of confidence the population mean is between 5.64 ft and 5.80 ft. 4. The population mean 5.72 0.08 feet at a 95% level of confidence. Confidence levels and confidence intervals provide a measure of how “good” a point estimate estimates a population parameter. 8.1 Estimating When is Known (Page 3 of 25) Confidence Interval for A c-percent confidence interval for the population mean is an interval computed from sample data in such a way that c is the probability of generating an interval containing the actual value of . That is, P(x E x E) c , Shaded Area = c xE xE x -axis The probability that is on this interval is c where E is the maximum margin of error when estimating with x. In words, P(x E x E) c means . . . 1. The probability that the population mean is between x E and x E is c. 2. The population mean is between x E and x E at a confidence level of c. 3. The population mean is E at a c-percent level of confidence. 5. If we repeat the experiment many times with the same sample size, then c proportion of the intervals calculated will contain the population mean . Thus, 1 c proportion of the intervals will not contain . 8.1 Estimating When is Known (Page 4 of 25) Example 1 Jackie has been jogging 2 miles a day for years and she records her times. A sample of 90 of these times has a mean of 15.60 minutes and a known standard deviation of 1.80 minutes. a. Find a 95% confidence interval for . Draw and label the normal distribution illustrating the confidence interval. Solve without using the ZInterval function (see below). b. Find E, maximum error in estimating with x at the confidence level c. c. Write the conclusion in probability notation. i.e. P(x E x E) c d. Summarize your conclusion in one sentence [relvant to the application]. 8.1 Estimating When is Known (Page 5 of 25) Using the TI-83/84 ZInterval Function The ZInterval (STAT / TESTS / 7: ZInterval) function computes a confidence interval for and unknown population mean when the population standard deviation is known. Input: Output: , x , and c-level STATS The interval from x E to x E , where 1 E (interval length) 2 Example 2 a. Compute the 95% confidence interval in example 1. Use the ZInterval function. b. Summarize your results in a complete sentence relevant to this application. At a 95% level of confidence, the population mean of all 2mi jogging times for Jackie is between 15.23 and 15.97 minutes. 8.1 Estimating When is Known (Page 6 of 25) Section 8.1 Homework Instructions Steps to find a c% confidence interval for 1. Sketch a normal curve illustrating the c% confidence interval for . Label x E , x E , and x . Where E is the margin of error when estimating with x at a confidence level of c. xE xE 2. Without using the ZInterval function, compute the c% confidence interval for the population mean. That is, x E invNorm(area to the left of x E, x , x ) x E invNorm(area to the left of x E, x , x ) Use the estimate x x , and x / n . 3. Find E, is the maximum error in estimating with x at a confidence level of c. It is computed as follows E = “half the interval length” from step 2 E 12 (x E) (x E) 4. Write the confidence interval in probability notation. i.e. P(x E x E) c 5. Summarize your results in a concise, complete sentence relevant to the problem. That is, At the c% confidence level the population mean of all ____________________ is between _____ and _____ [units]. x -axis 8.1 Estimating When is Known (Page 7 of 25) Guided Exercise 2 Jason jogs 3 miles per day and records his times. A sample of 90 of these times has a mean of 21.50 minutes and a known standard deviation of 2.11 minutes. Find the 99% confidence interval for the population mean by completing steps 1-5 above. a. Sketch a normal curve to illustrate the 99% confidence interval for the mean in his application. Label the axis. b. Without using the ZInterval function, find the 99% confidence interval for the population mean. c. Find E = “half the interval length” d. Write the confidence interval in probability notation. i.e. P(x E x E) c e. Summarize your results in a concise, complete sentence relevant to the problem. At a ____% confidence level, the mean of _______________ _________________________________________________ is between ____________ and _____________. 8.1 Estimating When is Known (Page 8 of 25) Guided Exercise 3 An automobile loan company wants to estimate the amount of the average car loan during the past year. A random sample of 200 loans had a mean of $8225 and a known standard deviation of $762. Find the 95% confidence interval for the population mean by completing steps 1-5 above. a. Sketch a normal curve to illustrate the 95% confidence interval for the mean in his application. Label the axis. b. Without using the ZInterval function, find the 95% confidence interval for the population mean. c. Find E = “half the interval length” d. Write the confidence interval in probability notation. i.e. P(x E x E) c e. Summarize your results in a concise, complete sentence relevant to the problem. At a ____% confidence level, the mean of _______________ _________________________________________________ is between ____________ and _____________. 8.1 Estimating When is Known (Page 9 of 25) 8.1 (was 8.4) Estimating the Sample Size n Critical Value z c is called the critical value for a confidence level c if P(zc z zc ) c That is, z c is the z-score such that the area under the standard normal curve between zc and z c is c. In words we say . . . a. “the probability that a randomly selected z-value is between zc and z c is c.” Or b. “at a c-percent level of confidence we can say that a randomly chosen z will be between zc and zc .” Shaded Area = c zc zc z-axis For Example 1. If c 0.90 , then P(z0.90 z z0.90 ) 0.90 . Compute z0.90 . 2. If c 0.95 , then P(z0.95 z z0.95 ) 0.95 . Compute z0.95 . 3. If c 0.99 , then P(z0.99 z z0.99 ) 0.99 . Compute z0.99 . Estimating Sample Size n for Estimating 8.1 Estimating When is Known (Page 10 of 25) If, with a confidence level of c, we want our point estimate x to be within E units of , then we choose the sample size n to be zc 2 n E , where z c is the critical value for a confidence level of c. Example 6 A sample of 50 salmon is caught and weighed. The sample standard deviation of the 50 weights is 2.15 lb. How large of a sample should be taken to be 97% confident that the sample mean is within 0.20 lb of the mean weight of the population? Find z c (to the nearest thousandth) and n. Then summarize your results in a complete sentence relevant to this application. 8.1 Estimating When is Known (Page 11 of 25) Example 7 An efficiency expert wants to determine the mean time it takes an employee to assemble a switch on an assembly line. A preliminary study of 45 observations found a sample standard deviation of 78 seconds. How many more observations are needed to be 92% certain that the mean of the sample will vary from the true mean by no more than 15 seconds? Find z c (to the nearest thousandth) and n. Then summarize your results in a complete sentence relevant to this application. Guided Exercise 6 The dean wants to estimate the average teaching experience (in years) of the faculty members. A preliminary random sample of 60 faculty yields a sample standard deviation of 3.4 years. How many more faculty should be sampled to be 99% confident that the sample mean does not differ from the true mean by more than 0.5 years? Find z c (to the nearest thousandth) and n. Then summarize your results in a complete sentence relevant to this application. 8.2 Estimating When is Unknown (Page 12 of 25) 8.2 Estimating When is Unknown When the population standard deviation is unknown, it is approximated by the sample standard deviation s. The TInterval function works with what is called the Student’s t-distribution where all statistical “fudge factors” necessary to accommodate approximating with s are built into the function. The TInterval function (TI-83: STAT / TESTS / 8: TInterval) Input: STATS DATA Output: The interval from x E to x E , where 1 E (interval length) 2 x, s, n, and c-level data list and c-level or Homework Instructions for Section 8.2 1. Omit exercises #1-4 2. When asked to find a confidence interval, do the following: a. Find the c% confidence interval for the mean . Write it in probability notation b. Summarize your results in a complete sentence relevant to the application. 8.2 Estimating When is Unknown (Page 13 of 25) Example 4 An archeologist discovered a new, but extinct, species of miniature horse. The only seven known samples show shoulder heights (in cm) of 45.3, 47.1, 44.2, 46.8, 46.5, 45.5, and 47.6. Find the 99% confidence interval for (the mean height of the entire population of ancient horses) and the error E. Then summarize your results in a complete sentence relevant to this application. a. Find the 99% confidence interval for the mean . Write it in probability notation b. Summarize your results in a complete sentence relevant to the application. Guided Exercise 3 A company produced a trial production run of 37 artificial sapphires. The mean weight is 6.75 carats and the standard deviation is 0.33 carats. Find the 95% confidence interval for the mean weight of all artificial sapphires and the error E. Then summarize your results in a complete sentence relevant to this application. 8.3 Estimating p in a Binomial Experiment (Page 14 of 25) 8.3 Estimating p in a Binomial Experiment Large Sample Size Assumption If np > 5 and nq > 5, then the sample size n is large enough so that the binomial distribution can be approximated by a normal distribution, and a c% confidence interval for p is expressed as P( pˆ E p pˆ E) c where pφ is the point estimate for p. TI-83 1-PropZInt function: STAT / TESTS / A: 1-PropZInt Input: x = r = number of successes n = number of trials c-level = confidence level Output: ( pˆ E, pˆ E), pˆ , n Where E (the maximum error in using pφ as a point estimate for p for the given confidence level) is one-half the interval length. 8.3 Estimating p in a Binomial Experiment (Page 15 of 25) Example 5 Suppose 800 students were given flu shots and 600 did not get the flu. Assuming all 800 were exposed to the flu: a. What is S, n, and r (note: r is S= input as variable x on the TI-83)? r= n= b. What are the point estimates for p and q (i.e. pφ and qφ)? p̂ q̂ c. Is n large enough to approximate the binomial distribution with a normal distribution? Why? np̂ d. Find the 99% confidence interval for p. P( p̂ E p p̂ E) 0.99 e. Summarize your results in a complete sentence relevant to this application. nq̂ 8.3 Estimating p in a Binomial Experiment (Page 16 of 25) Guided Exercise 4 A random sample of 195 books at a bookstore showed that 68 of the books were nonfiction. a. Find S and pφ. b. Is the sample size large enough to approximate a normal distribution with a binomial distribution? Why? c. Find the 90% confidence interval for p to the nearest thousandth (3 decimal places). d. Summarize your results in a complete sentence relevant to this application. Homework Instructions for Section 8.3 Problems When asked to find the c% confidence interval for p, do the following four steps. 1. Find S and pφ 2. Determine if the sample size is large enough to approximate a normal distribution with a binomial distribution? 3. Find the c% confidence interval for p to the nearest thousandth (3 decimal places). 4. Summarize your results in a complete sentence relevant to this application. 8.3 Estimating p in a Binomial Experiment (Page 17 of 25) A Margin of Error, E, is the maximum error when using a point estimate for a population parameter at a given confidence level. General Interpretation of Poll Results 1. When a poll states the results of a survey, the proportion reported is pφ (the sample estimate of the population proportion). 2. The margin of error is the maximal error E of a [95%, usually] confidence interval for p. 3. If pφ is obtained from a poll, Then a 95% confidence interval for the population proportion p is pφ E p pφ E . Guided Exercise 5 A random sample of 315 households were surveyed. Chances are 19 of 20 that if all adults had been surveyed, the findings would differ from the poll results by no more than 2.6% in either direction. One question was asked: “Which party would do a better job handling education?” The possible responses were Democrats, Republicans, neither, or both. The poll reported that 32% responded Democrat. a. What confidence level corresponds to the phrase “chances are 19 of 20 that if . . . .” b. What is S, n, and the sample statistic pφ for the proportion responding Democrat? c. Find E. Find the 95% confidence interval for p those who would respond Democrat. d. Summarize your results in a complete sentence relevant to this application. 8.3 Estimating p in a Binomial Experiment (Page 18 of 25) 8.3 Estimating Sample Size n for Estimating p (a) If, with a confidence level of c, we want our point estimate pφ to be within E units of p, then we choose the sample size n to be z n pφ qφ c E 2 where z c is the z-score corresponding to a confidence level of c. (b) If no estimate for p is available, we can say with a confidence level of at least c that the point estimate pφ will be within E units of p by choosing zc 2 n 0.25 E Example 8 A buyer for a popcorn company wants to estimate the probability p that a kernel purchased from a particular farm will pop. Suppose a random sample of n kernels is taken and r of these kernels pop. The buyer wants to be 95% certain that the point estimate pφ will be within 0.01 units of p. a. Find z c and E. b. If no estimate for p is available, how large a sample should the buyer use? (i.e. how large should n be)? c. A preliminary study showed that p was approximately 0.86. Now, how large a sample should be used? 8.3 Estimating p in a Binomial Experiment (Page 19 of 25) Guided Exercise 7 The health department wants to estimate the proportion of children who require corrective lenses for their vision. They want to be 99% sure that the point estimate for p will have a maximum error of 0.03. a. If no other information is known, find E and z c . Estimate the sample size required. b. Suppose a preliminary random sample of 100 children indicates that 23 require corrective lenses. How large should n be? 8.4 Estimating 1 2 and p1 p2 (Page 20 of 25) 8.4 Estimating 1 2 and p1 p2 Independent and Dependent Samples In order to make a statistical estimate about the difference between two populations, we need to have a sample from each population. Two samples are independent if the sample from one population is unrelated to the sample from the other. However, if each measurement in one sample can be naturally paired with measurements of another sample, the two samples are said to be dependent (such as before and after samples). Guided Exercise 8 Classify the pairs of samples as dependent or independent. a. In a medical experiment, one group is given a treatment and another group is given a placebo. After a period of time both groups are measured for the same condition. b. A group of Math students is given a test at the beginning of a course and the same group is given the same test at the end of the course. 8.4 Estimating 1 2 and p1 p2 (Page 21 of 25) Theorem 8.1 Let x1 and x2 have normal distributions. If we take independent random samples of size n1 from x1 and n2 from x2 , then the variable x1 x2 has 1. a normal distribution 2. a mean of 1 2 3. a standard deviation of 12 n1 22 n2 Estimating 1 2 When 1 and 1 are Known A c% confidence interval for 1 2 is expressed as (x1 x 2 ) E 1 2 (x1 x2 ) E This interval is the output of the TI-83 function 2-SampZInt. TI-83function 2-SampZInt (STAT / TESTS / 9: 2-SampZInt) 1, 2 , x1, n1, x2 , n2 , c level Input: Output: Interval from (x1 x 2 ) E to (x1 x2 ) E Where E is one half the interval length output by the 2-SampZInt function. 8.4 Estimating 1 2 and p1 p2 (Page 22 of 25) Example 9 Suppose a biologist is studying data from Yellowstone streams before and after a 1988 fire. A random sample of 167 fishing reports in the years before the fire showed the average catch per day of 5.2 trout with 1.9 trout. After the fire a sample of 125 fishing reports showed the average catch per day of 6.8 trout with 2.3 trout. a. Are the sample independent? b. Compute a 95% C.I. for 1 2 . At a 95% level of confidence ________ < 1 2 < _______. c. Explain the meaning of part b. Estimating 1 2 When 1 and 1 are Unknown A c% confidence interval for 1 2 is expressed as (x1 x 2 ) E 1 2 (x1 x2 ) E This interval is the output of the TI-83 function 2-SampTInt. TI-83function 2-SampTInt (STAT / TESTS / 0: 2-SampTInt) x1 , s1 , n1 , x2 , s2 , n2 , c-level, pooled: yes Input: Output: Interval from (x1 x 2 ) E to (x1 x2 ) E Where E is one half the interval length output by the 2-SampTInt function. 8.4 Estimating 1 2 and p1 p2 (Page 23 of 25) Example 10 Suppose that a random sample of 29 college students was divided into two groups. The first group had 15 people and was given 1/2 liter of red wine before going to sleep. The second group of 14 people was not given alcohol before going to sleep. Both groups went to sleep at 11 p.m. The average brain wave activity (in hertz) between 4 and 6 a.m. was measured for each participant. The results follow: Group 1 16.0 19.6 19.9 20.9 20.3 20.1 16.4 20.6 20.1 22.3 18.8 19.1 17.4 21.1 22.1 x1 19.65 hz , s1 1.86 hz Group 2 8.2 7.6 5.4 6.8 10.2 6.4 6.5 8.8 4.7 5.4 5.9 8.3 2.9 5.1 x2 6.59 hz , s2 1.91 hz a. Are the groups independent? b. Compute the 90% C.I. for 1 2 and write it in probability notation. c. Summarize the results of part b in a single sentence relevant to this application. 8.4 Estimating 1 2 and p1 p2 (Page 24 of 25) Guided Exercise 9 a. A study reported a 90% confidence interval for the difference of the means to be 10 1 2 20 . What can you conclude about the values of 1 and 2 . b. A study reported a 95% confidence interval for the difference of proportions to be 0.32 p1 p2 0.16 . What can you conclude about the values of p1 and p2 . 8.4 Estimating 1 2 and p1 p2 (Page 25 of 25) Confidence Interval for p1 p2 (Large Samples) If n1 pφ1 5 , n1qφ1 5 , n2 pφ2 5 and n2 qφ2 5 , then the c% confidence interval for p1 p2 is expressed as ( pφ1 pφ2 ) E p1 p2 ( pφ1 pφ2 ) E where E is the maximum error in using pφ1 pφ2 as an estimate for p1 p2 at a c% confidence level. TI-83 function 2-PropZInt (STAT / TESTS / B: 2-PropZInt) r1 x1 , n1 , r2 x2 , n2 , c-level Input: Output: Interval from ( pφ1 pφ2 ) E to ( pφ1 pφ2 ) E Where E is one half the interval length. Exercise 14 The burn center at Community hospital is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. Of this group, 94 had no visible scars. Let p1 be the proportion of patients who received the plasma compress treatment and had no visible scars after treatment. Let p2 be the proportion of patients who did not receive the plasma compress treatment but still had no visible scars. a. Find the 95% confidence interval for p1 p2 . b. Summarize the results in a single sentence relevant to this application.