Chapter 8 - Notes

8.1 – Introduction to Confidence Intervals [VIDEO #1] Confidence Interval : An estimate found by using a ________________ ____________________ and ________________ and ____________________ an amount corresponding to how ___________________ we are that the interval created captures the population parameter of interest. - Higher confidence levels mean (circle one) [ wider or narrower ] intervals. Interpretation of a Confidence Level : A X% confidence level means that if we did many, many random samples and created X% confidence intervals for all of them, then we expect _________________ ____% of all those X% confidence intervals (CI’s) would capture the ______ ________________________. Approx. ________% of all those CI’s would _______ capture the pop. parameter due to unfortunate ______________________ error. Unless you somehow figure out the population parameter, you ______ ____ know if you CI actually captured it! Being 100% confident means you would have to include _________________________________________ _______________________________________________________________________________________. For example, I am 100% confident that the mean score on the next AP Stats test will be between __________. 3

What CI’s Say DIRECTIONS : Identify each statement as correct or incorrect. If you believe a statement to be incorrect, explain exactly why. 1. 95% of the time, the number of hours people sleep is between 6.55 and 8.25. 2. 95% of the confidence intervals constructed by this method will contain the true mean number of hours that adults sleep at night. 3. We are 95% confident that the true mean number of hours that adults sleep at night is between 6.55 and 8.25. 4. 95% of the confidence intervals we construct will give us the interval (6.55, 8.25). 5. There is a 95% probability that the mean number of hours that adults sleep is between 6.55 and 8.25. 6. If 100 random samples of the given size are picked and a 95% confidence interval is calculated from each, then the true mean number of hours that adults sleep will be in approximately 95 of the resulting intervals. 7. We are 95% certain that the true mean number of hours that adults sleep is 7.4 with a margin of error of 0.85. 8. This 95% confidence interval is an interval computed from sample data by a method that has a 95% probability of producing an interval containing the true mean number of hours that adults sleep. 9. This confidence interval consists of plausible values for the true mean number of hours that adults sleep. Associated with the interval is a 95% level, which measures one’s confidence that the true mean number of hours that adults sleep lies within 6.55 and 8.25. 10. “To be more specific, the laws of probability say that if Gallup was to conduct the same survey 100 times, asking people in each survey to rate the job President X is doing as president, in 95 out of those 100 polls, his rating would be between 46% and 54%. In five of those surveys, his rating would be higher or lower than that due to chance error.” (From www.gallup.com ) 11. "In theory, in 19 cases out of 20, overall results based on such samples will differ by no more than 3 percentage points in either direction from what would have been obtained by seeking to interview all American adults." (From The New York Times ) 6

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8.2 – Introduction to CI’s for one proportion [ VIDEO #3] Earlier, we discussed about the definition of a confidence interval being a point estimator (a statistic, ¯ x or ^ p ) plus or minus a margin of error related to how confident we were about capturing the population parameter of interest (µ or p ). The formal definition is this: ___________________  (________________________) · (________________________________________) The part after the plus/minus is our margin of error that can be broken down into two parts. The “critical value” is related to the “how confident” we want to be with our interval while the “standard error of the statistic” is just our standard deviation for the sampling distribution of ¯ x or ^ p we used last chapter. Confidence Interval for One Proportion Statistic  (Critical Value)(Standard Error of the Statistic) Turns specifically into… ^ p  z ¿ √ ^ p ( 1 − ^ p ) n Conditions: 1. The data must come from a __________________________________ from the population of interest. a. YOU MUST WRITE BEFORE DOING ANY CALCULATIONS: “We have a SRS of (size/#) (people or whatever you sampled) to represent ALL (people or whatever you sampled) . b. If there is no mention of a random sample, then write “Sample not known to be random, proceed with caution!” and continue with the problem. 2. The observations of your sample must be _____________________________. a. Check the “10%” condition a.k.a. “Population > 10 · n ”. SHOW YOUR WORK! If it checks, then write “We may proceed to calculate the standard error.” b. If NOT, then you need to use the Finite Population Correction formula we mentioned before but will not use in this class, so write “Standard error cannot be safely calculated.” STOP. 3. The sampling distribution of ^ p must be _____________________ ________________________. a. Check the condition n · ^ p > 10 and n (1 - ^ p ) > 10. SHOW THE ACTUAL WORK HERE! If it checks, then write “We may assume the sampling distribution is approx. normal.” b. If NOT, then write “Sampling distribution not normal, CI cannot be calculated.” STOP. c. NOTE : The CLT does NOT work with proportion problems, so don’t use it here!!! 12

8.2 - Choosing the Sample Size [ VIDEO #5] In planning a study, we may want to choose a sample size that will allow us to estimate the parameter within a given margin of error. m = z ¿ √ ^ p ( 1 −^ p ) n Problem: Until we actually do the study we won’t know what ^ p is. We need to guess approximately what ^ p will be. We will call this guess p * . Remedy #1: Guess p * based on past experience Remedy #2: Use p * = .5. The margin of error is largest when ^ p = .5, so this guess is conservative in the sense that if we get any other ^ p value when we do our study, we will get a margin of error smaller than planned. Use p * = .5 when you expect the true value of p to be roughly between .3 and .7 Finding Sample Size Gloria Chavez and Ronald Flynn are candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. You will contact a SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or .03. How large of a sample do you need? 15

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You also have the t-distribution values in the back of your formula sheet. If you have a TI-83, ask me for the “invT” program, so you can get the values on your calculator. If you have a TI-84, “invT” is found below “normalcdf” and “invNorm”. Practice Problems: 1. The scores of four roommates on the Law School Aptitude Test have mean ¯ x =589 and standard deviation s = 37. What is the standard error of the mean? 2. What critical value t * from Table B satisfies each of the following conditions? a. The t distribution with 10 degrees of freedom and has probability 0.01 to the right of t * . b. The t distribution with 28 degrees of freedom and has probability 0.90 to the left of t * . c. The one-sample t statistic from a sample of 42 observations that has probability 0.05 to the right of t * . d. The one-sample t statistic from a SRS of 980 observations and has probability 0.975 to the left. e. A 95% CI based on a SRS of 15 observations. f. A 99% CI from a sample size of 110 observations. g. A 90% CI based on 30 observations. 21

8.3 – Calculating CI’s for One Mean [ VIDEO #7] Confidence Interval for One Mean Statistic  (Critical Value)(Standard Error of the Statistic) Turns specifically into… ¯ x  t ¿ s √ n Conditions: 1. The data must come from a __________________________________ from the population of interest. a. YOU MUST WRITE BEFORE DOING ANY CALCULATIONS: “We have a SRS of (size/#) (people or whatever you sampled) to represent ALL (people or whatever you sampled) . b. If there is no mention of a random sample, then write “Sample not known to be random, proceed with caution!” and continue with the problem. 2. The observations of your sample must be _____________________________. a. Check the “10%” condition a.k.a. “Population > 10 · n ”. SHOW YOUR WORK! If it checks, then write “We may proceed to calculate the standard error.” b. If NOT, then you need to use the Finite Population Correction formula we mentioned before but will not use in this class, so write “Standard error cannot be safely calculated.” STOP. 3. The sampling distribution of ¯ x must be _____________________ ________________________. a. If you know σ somehow, then you can say and use the normal distribution (z-scores). If you do not know σ, then you can saw and use the t-distribution (t-scores). ( More likely ) b. If the population is known to be Normal ( REGARDLESS OF SAMPLE SIZE ), then write “Since the population is known to be normal, then we may assume the sampling distribution can follow an (select one: approx. normal or t ) distribution.” c. If NOT, then check for a sufficiently large enough sample size (> 30) for the CLT to apply If so, write “Sampling distribution can use t-procedures due to the CLT and sample size.” d. If sample size is less than 30: t procedures can be used except in the presence of outliers or strong skewness (some skewness OK). Plot your data (stemplot or boxplot) to check for Normality (roughly symmetric, single peak, no outliers). If the data checks out, then write “Based on the data shown, we may assume a t- distribution with ___ df.” 22

e. If the data are STRONGLY skewed or if outliers are present, do not use t . STOP . Ex 1 : A company manufactures portable music devices—called “mBoxes”—for the listening pleasure of on- the-go teens. The mBox uses batteries that are advertised to provide, on average, 25 hours of continuous use. The students in Mr. Jones’s statistics class—looking for any excuse to listen to music while doing their homework—decide to use their statistics to test this advertising claim. To do this, new batteries are installed in eight randomly selected mBoxes, and they are used only in Mr. Jones’s class until the batteries run down. Here are the results for the lifetimes of the batteries (in hours): 15 22 26 25 21 27 18 22 Construct and interpret a 95% CI for the mean number of hours (µ) that the batteries can be expected to last. C: Check the Conditions of the Inference Procedure C: Carry out the Inference Procedure - Calculations! Confidence Interval Formula: Statistic  (Critical Value)(Standard Error of the Statistic) . ¯ x ± t ¿ s √ n C: Conclusion: Interpret the Interval 23

I am _____% confident that the mean _____________________ for all __________________ is between ____________ and _____________. 8.3 – Calculating a Confidence Interval for a Mean Difference (Matched Pairs) [ VIDEO #8] Examples of Matched Pairs Experimental Designs: 1. Two seeds of the same type of plant are planted in the same pot. One is randomly selected and treated with a new fertilizer. The other seed serves as the control. We want to estimate the mean difference in growth for 10 such pairs of plants. 2. An elementary school teacher uses a matched pairs design when giving a pretest and posttest and then determining if the students improved. 3. A study tries to determine whether caffeine addicts are more depressed when they receive a “caffeine placebo” rather than a “caffeine pill”. What makes an experimental design “Matched Pairs”? _______________________________________ ___________________________________________________________________________________ How do you analyze a matched pairs data set? _____________________________________________ ___________________________________________________________________________________ Matched Pairs Procedure Example: Our subjects are 11 people diagnosed as being dependent on caffeine. Each subject was barred from coffee, colas, and other substances containing caffeine. Instead, they took capsules containing their normal caffeine intake. During a different time period, they took placebo capsules. The order in which subjects took caffeine and the placebo was randomized. One of the tests given to the subjects was “Depression” as scored by the Beck Depression Inventory. Higher scores show more symptoms of depression. Construct and interpret a 90% confidence interval for the mean change in depression score.  Conditions :  Calculations : 24

 Conclusion : 25

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a) Calculate the standard error of the mean. 28

Ch 8 Multiple Choice Review Problems 1. The weights (in pounds) of three adult males are 160, 215, and 195. The standard error of the mean of these three weights is A) 190. B) 27.84. C) 22.73. D) 16.07. E) 13.13. 2. Scores on the Math SAT (SAT-M) are believed to be normally distributed with mean  . The scores of a random sample of three students who recently took the exam are 550, 620, and 480. A 95% confidence interval for  based on these data is A) 550.00 ± 173.88. B) 550.00 ± 142.00. C) 550.00 ± 128.58. D) 550.00 ± 105.01. E) 550.00 ± 79.21. 3. You are told that the sample proportion ^ p of those who phoned in and answered yes is ^ p = 0.70 and the standard error of the sample proportion is 0.0459. The number of people who phoned in A) Is 21. B) Is 50. C) Is 100. D) Is 200. E) Cannot be determined from the information given. 4. An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment with major defects prior to using the potatoes to make potato chips. She intends to compute a 95% confidence interval for p . To do so, she selects an SRS of 50 potatoes from a shipment of over 2000 potatoes on a truck. Suppose that only two of the potatoes sampled are found to have major defects. Which of the following assumptions for inference about a proportion using a confidence interval are violated? A) n is so large that both n ^ p < 10 and n (1 – ^ p ) < 10. B) n is so large that both the count of successes n ^ p and the count of failures n (1 – ^ p ) are 10 or more. C) The population size is too small. D) The population is at least 10 times as large as the sample. E) There appear to be no violations. 5. Which of the following is an example of a matched-pairs design? A) A teacher compares the pretest and posttest scores of students. B) A teacher compares the mean score of one class of 30 students on a standardized test with the mean score of another class of 30 students. 29

C) A teacher compares the scores of students in her class on a standardized test with the national average score. D) A teacher calculates the average of scores of students on a pair of standardized tests and wishes to see if this average is larger than 80%. E) A teacher compares the standardized test scores of students using a computer-based method of instruction with the scores of other students using a traditional method of instruction. 6. An advertiser wishes to see if a new advertisement is effective in promoting an existing product. The previous advertisement has a recognition score of 3.7. An SRS of 23 potential buyers resulted in a recognition score of 3.4 for the sample. The standard deviation of the population is known to be 1.7. Which of the following required conditions for conducting a z-test has not been met? A) The population is at least 10 times the sample size. B) The data are taken from a simple random sample. C) The data appear to be approximately normal. D) The decision of each buyer is independent. E) All of the required conditions are met. 7. Suppose we want to compute a 90% confidence interval for the average amount  spent on books by freshmen in their first year at a major university. The interval is to have a margin of error of $2, and the amount spent has a normal distribution with standard deviation  = $30. The number of observations required is closest to A) 25. B) 30. C) 608. D) 609. E) 865. 8. Other things being equal, the margin of error of a confidence interval increases as A) The sample size increases. B) The sample mean increases. C) The population standard deviation increases. D) The confidence level decreases. E) None of the above. 9. A 95% confidence interval for the mean  of a population is computed from a random sample and found to be 9 ± 3. We may conclude that A) There is a 95% probability that  is between 6 and 12. B) 95% of values sampled are between 6 and 12. C) If we took many, many additional random samples and from each computed a 95% confidence interval for  , approximately 95% of these intervals would contain  . D) There is a 95% probability that the true mean is 9 and a 95% chance that the true margin of error is 3. E) All of the above are true. A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a random sample (assume it is an SRS) of 1200 registered voters and found that 620 would vote for the Republican candidate. Let p represent the proportion of registered voters in the state that would vote for the Republican candidate. 30

10. Referring to the information above, a 90% confidence interval for p is A) 0.517 ± 0.014 B) 0.517 ± 0.022 C) 0.517 ± 0.024 D) 0.517 ± 0.028 E) 0.517 ± 0.249 Preparing for the Confidence Interval Test For the Chapter 8 Test, Need To Know How To:  Find a z* value for any confidence level (for the calculation of a CI) using Table A or on your calculator. For example, what z* corresponds to 82% confidence?  Find the required sample size needed to obtain a specific margin of error. PS – To do this you need to know the formula for margin of error.  Interpret a given confidence interval. If we took many SRS and calculated CI’s about 95% of them would capture the true population parameter (5% of them would miss the population parameter)  Explain how a change in the sample size or confidence level affects the length of a CI.  Construct and Interpret a 90%, 95%, or 99% CI for a given data set or for given information.  Make sure you know your formulas and don’t get the components of them mixed up. You should know them off by heart from doing HW problems.  Check the conditions for each procedure, including the fact that outliers and/or strong skewness invalidate t procedure results  Calculate the standard error of the mean  Find t * with either Table B or on your calculator.  The difference between a one sample t and a matched pairs t procedure  Verify that the population is normal with either a stemplot, histogram, or boxplot, be sure to note the absence of outliers and strong skewness  Calculate the standard error for a one sample proportion.  Find the necessary sample size for a specific margin of error or confidence interval length (A CI of length .5 has a margin of error of .25)  Calculate a confidence interval for a one sample proportion for any specified confidence level 31