Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Estimating the Value of a Population Parameter 9

Chap 22

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Estimating a Population Proportion 9.1

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A point estimate is the value of a sample statistic that estimates the value of a population parameter. The point estimate for the population proportion is where “x” is the number of “successes” (you determine what is a success) and “n” is the sample size. 9-4

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. In 2008, a University Poll asked 1783 registered voters nationwide whether they favored or opposed the death penalty for persons convicted of murder were in favor (“success”). Based on this sample, obtain a point estimate for the proportion of ALL registered voters (pop) who are in favor of the death penalty for persons convicted of murder. 9-5 Calculating a Point Estimate for the Population Proportion

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A confidence interval for a pop parameter consists of an interval about a point estimate. The level of confidence represents the probability that this interval will actually contain the population parameter we are trying to estimate. The “ level of confidence” we specify is denoted “c” or “1 - α” 9-6

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Another way to look at Confidence Intervals 95% level of confidence (“c” = 0.95 or “α” = 0.05) means that if we construct 100 different confidence intervals, each based on a different sample from the same population, then 95 of those intervals will actually contain the pop parameter we are trying to estimate, and 5 will not. 9-7

8 The relation between “c” and “α” c + α = 1 c + α = 1 c = 1 – α α = 1 – c c = 1 – α α = 1 – c So, if “c” = 0.90 for 90% Conf Interval, then α = 1 – c = 0.10

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Confidence interval estimates for the population proportion Interval = point estimate ± margin of error “E” The margin of error of the interval is a measure of how much confidence we have that the interval actually contains the pop parameter we are trying to estimate, based on our sample (point estimate). 9-9

Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 Level of Confidence “p” The level of confidence “c” is the probability that the interval estimate actually contains the true population proportion “p”. z z = 0 zczc zczc Critical values (1 – c)/2 c is the area beneath the normal curve between the critical values. The remaining area in both tails is (1 – c) c Use Table 5 or TI-84 to find the corresponding z - scores.

Larson & Farber, Elementary Statistics: Picturing the World, 3e 11 Common Levels of Confidence If our level of confidence is c = 0.90, then we are 90% confident that the interval estimate will contain the true population proportion “p”. z z = 0 zczc zczc The corresponding z-crit scores are ± See bottom of Table 5 for some commonly used c-values. c =  z c =  z c = 1.645

Larson & Farber, Elementary Statistics: Picturing the World, 3e 12 Point Estimate for Population p In a sample survey of 1250 adults, 450 of them said that their favorite sport to watch is baseball. Find a point estimate for the population proportion of adults who say their favorite sport to watch is baseball. The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%. (q-hat is = 0.64 or 64%) n = 1250x = 450

Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Confidence Intervals for p A c - confidence interval for the population proportion “p” is “margin-of-error” The probability that this confidence interval actually contains the population “p” is “c”. Construct a 90% confidence interval for the proportion of all adults who say baseball is their favorite sport to watch. n = 1250x = 450

Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 Confidence Intervals for p Based on our sample, we can say with 90% confidence that the proportion of all adults (pop) who say baseball is their favorite sport to watch is between 33.8% and 38.2%. Left end = Right end = n = 1250 x = 450

Chap 6 15 TI-84 Conf Interval (Proportions) 450/1250 successes c = /1250 successes c = 0.90 p-hat = 0.36 q-hat = 0.64 p-hat = 0.36 q-hat = 0.64 npq = (1250)(0.36)(0.64) = 288 > 10 npq = (1250)(0.36)(0.64) = 288 > 10 (approx normal, so can use z-scores!) (approx normal, so can use z-scores!) 1-PropZInt Test: x=450 n=1250 c=0.90 Ans: (0.338, 0.382) Exam Q  E=???

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. For a random sample of size “n”, the sampling distribution of is approximately normal with mean standard deviation IFF, distribution is normal: npq ≥ 10 (recall: q = 1-p) 9-16 Sampling Distribution of

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The margin of error, E, in a confidence interval for a population proportion is given by: 9-17 Margin of Error Ex: if c =.90, then Zcrit = Note: n must be ≤ 0.05N to construct a valid interval.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Interpretation of a Confidence Interval A 90% confidence interval indicates that: 1) 90% of all random proportion samples of size “n” taken from the population will lie within that interval OR 2) that the probability that our interval actually contains the pop parameter is

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. In 2008, a University Poll asked 1783 registered voters whether they favored/opposed the death penalty for persons convicted of murder were in favor (“success”). 1123/1783 = 0.63 Obtain a 90% confidence interval for the proportion of ALL registered voters (pop) who are in favor of the death penalty for persons convicted of murder Constructing a Confidence Interval for a Population Proportion

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.   So, distribution is normal and n < 0.05N (assumed)  C = 0.90 or α = 0.10 so z crit =  Low Interval bound:  Upper interval bound: 9-20 Solution

Chap 6 21 TI-84 Conf Interval (Proportions) 1123/1783 successes c = /1783 successes c = 0.90 Stat:Tests:A:1-PropZInt Test: x=1123 n=1783 c=0.90 x=1123 n=1783 c=0.90 Ans: (0.6110, ) Margin of Error “E”=??? E = ( )/2 = or 1.88% E = ( )/2 = or 1.88%

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 90% Conf. Interval: (0.6110, ) Based on our sample, we are 90% confident that the proportion of ALL registered voters (pop) who are in favor of the death penalty for those convicted of murder is between and ( between 61.10% and 64.86%) 9-22 Conclusion

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The sample size required to obtain that confidence for our p interval, with a margin of error E is: where E is normally in % because we are dealing in proportions Sample Size Needed for Estimating the Population Proportion p

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. If you have not yet done the survey, but want an estimate for a minimum sample size, then use : 9-24 Sample Size Needed for Estimating the Population Proportion p

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A sociologist wanted to determine the current percentage of US residents that speak only English at home. What minimum sample size should she use if she wants her max estimate error to be 3% (E), with 90% confidence, assuming she uses the 2000 Census Survey result of 82.4% as a preliminary estimate? 9-25 Determining Sample Size

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.  E = 0.03  Round this up to 437 randomly selected American residents which is the min sample size for a 90% confidence that the max interval error is 3% 9-26 Solution

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Estimating a Population Mean 9.2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The Margin of Error “E” depends on three factors: Level of confidence: As the level of confidence increases, the margin of error widens. Sample size: As the size of the random sample increases, the margin of error shrinks. Population standard deviation “ σ” : The more spread there is in the population, the wider our margin of error will be for a given level of confidence. 9-28

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Pennies minted after 1982 are made from 97.5% zinc and 2.5% copper. The following data represent the weights (in grams) of 17 randomly selected pennies minted after Based on this sample, create a point estimate for the population mean weight of all pennies minted after Computing a Point Estimate for a Mean

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The sample mean is: Our sample mean forms a “point estimate” of the pop mean weight μ of all pennies which is grams. 9-30

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. If the population from which a sample size “n<30 ” is drawn follows a normal distribution, the distribution of follows Student’s t-distribution with (n – 1) degrees of freedom where is the sample mean and “s” is the sample standard deviation “Student’s t-Distribution”

9-32 Histogram for z

9-33 Histogram for t

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 1.The t-distribution is different for different degrees of freedom. 2.As the sample size n increases, the distribution curve of “t” gets approximates the standard normal curve. 3. The area under the curve is 1. The area under the curve is symmetric to the right and left of As t increases or decreases without bound, the graph approaches, but never equals, zero. 9-34

Larson & Farber, Elementary Statistics: Picturing the World, 3e 35 The Student t - Distribution 3.As the # of degrees of freedom (d.f.) increase, the t-distribution approaches the normal distribution. 4.Above 30 d.f., the t-distribution is close to the standard normal z-distribution. t 0 Standard normal curve The tails in the t-distribution are “thicker” (further from the horizontal axis) than those in the standard normal distribution, so the area under the curve at large std devs is much greater. d.f. = 5 d.f. = 2

9-36 SEE Table VI !!

Larson & Farber, Elementary Statistics: Picturing the World, 3e 37 Critical Values of t Find the critical value t c for a 95% confidence interval when the sample size is 5. 95% of the area under the t-distribution curve with 4 degrees of freedom lies between t = ± t  t c =  t c = c = 0.95 alpha = 0.025

9-38 The figure to the left shows the graph of the t-distribution with 10 degrees of freedom. The area under the curve to the right of t is shaded. See Table VI: the value of t 0.20 with 10 degrees of freedom is This is the t-crit for a Confidence Interval of 60% and a sample size: n = 11.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A point estimate is the value of a sample statistic that estimates the value of a population parameter. For example, the sample mean is the point estimate for the population mean μ. 9-39

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Confidence Interval for μ  Data come from a simple random sample or randomized experiment.  Sample size is small relative to the population size or: n ≤ 0.05N  The data come from a population that is normally distributed, or the sample size is large Lower/Upper bound: Using and (n – 1) d.f. 9-40

Larson & Farber, Elementary Statistics: Picturing the World, 3e 41 Constructing a Confidence Interval In a random sample of 20 customers at a local junk food restaurant, the mean waiting time to order is 95 seconds, and the standard deviation is 21 seconds. Assume the wait times are normally distributed. Construct a 90% confidence interval for the mean wait time of all (pop) customers. So, based on this sample, we are 90% confident that the mean wait time for all customers is between 86.9 and seconds. = 95s = 21 t c = +/ n = 20 d.f. = 19 ± E = 95 ± sec < μ < sec = 8.1 sec

9-42 Using this sample of 17 pennies, construct a 99% confidence interval about the population mean weight (grams) of pennies minted after Use = 2.464g and s = 0.02g Constructing a Confidence Interval

9-43 Weight (in grams) of Pennies

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.   Lower bound: = – =  Upper bound: = = Based on this sample, w e are 99% confident that the mean weight of pennies minted after 1982 is between 2.45 and 2.48 grams. 9-44

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The sample size required to estimate the population mean, µ, with a level of confidence “c” with a specified margin of error, E, is given by where “n” is rounded up to the nearest whole number. 9-45

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. How large a sample of pennies would be required to estimate the mean weight of a penny manufactured after 1982 with a max error of grams with 99% confidence? Assume: s = 0.02g 9-46 Determining the Sample Size

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.  c=99%  α=1% so  s = 0.02  E = Rounding up, we find min sample size is: n =

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Estimating a Population Standard Deviation 9.3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. If a simple random sample of size n is obtained from a normally distributed population with mean μ and standard deviation σ, then has a chi-square distribution with (n-1) degrees of freedom. 9-49

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 1. It is not symmetric (skewed right). 2. The shape depends on the degrees of freedom (d.f.), just like the Student’s t- distribution. 3. As the number of d.f. increases, the chi- square distribution becomes more nearly symmetric(normal). 4. The values of χ 2 are always nonnegative (≥ 0) Characteristics of the Chi-Square Distribution

9-51

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Find the chi-square values that separate the middle 95% of the distribution from the 2.5% in each tail. Assume 18 degrees of freedom (d.f.) Finding Critical Values for the Chi-Square Distribution: TABLE VII

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Find the chi-square values that separate the middle 95% of the distribution from the 2.5% in each tail. Assume 18 degrees of freedom. χ = χ =

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. If a simple random sample of size n is taken from a normal population with mean μ and standard deviation σ, then a c% confidence interval for χ 2 is: Lower bound: Upper bound: Note: To find a (1-  )·100% confidence interval about “σ”, take the square root of the lower bound and upper bound Confidence Interval for χ 2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. One way to measure stock risk (volatility) is through the standard deviation of several stock prices. The following data represent the weekly gain/loss (%) of Microsoft stock for 15 randomly selected weeks. Compute the 90% confidence interval for the std dev (volatility/risk) of Microsoft stock – – – –3.90 – –1.61 – Constructing a Confidence Interval for a Population Variance and Standard Deviation

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. (Note: the data is approximately normal with no outliers) = % s = % s 2 = Table VII: χ = χ = for 14 d.f.  Lower variance bound:  Upper variance bound: We are 90% confident that the population stock variance interval is (13.04, 47.01), so the stock standard deviation (volatility) interval is (3.61%,6.86%) 9-56

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Which Procedure Do I Use? 9.4

58Chap 6 This next slide is very important. You should print it and study it….

Larson & Farber, Elementary Statistics: Picturing the World, 3e 59 To Use Normal or t-Distribution? Is n  30? Is the population normally, or approximately normally, distributed? You cannot use the normal distribution or the t-distribution. No Yes Is  known? No Use the normal distribution with If  is unknown, use s instead. YesNo Use the normal distribution with Yes Use the t - distribution with and n – 1 degrees of freedom.

Larson & Farber, Elementary Statistics: Picturing the World, 3e 60 To Use Normal or t-Distribution? Determine whether to use the normal distribution, the t-distribution, or neither. a.) n = 50, the distribution is skewed, s = 2.5 The normal distribution (z-interval) would be used because the sample size is > 30. b.) n = 25, the distribution is skewed, s = 52.9 Neither distribution would be used because n < 30 and the distribution is skewed (not normal) so cannot use t or z-interval. c.) n = 25, the distribution is normal,  = 4.12 The normal distribution (z-interval) would be used because although n < 30, the population is normal and pop standard deviation is known.

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Chap 262