Chapter 211 What Is a Confidence Interval?

Chapter 212 Thought Question 1 Suppose that 40% of a certain population favor the use of nuclear power for energy. (a)If you randomly sample 10 people from this population, will exactly four (40%) of them be in favor of the use of nuclear power? Would you be surprised if only two (20%) of them are in favor? How about if none of the sample are in favor?

Chapter 213 Thought Question 2 (b)Now suppose you randomly sample 1000 people from this population. Will exactly 400 (40%) of them be in favor of the use of nuclear power? Would you be surprised if only 200 (20%) of them are in favor? How about if none of the sample are in favor? Suppose that 40% of a certain population favor the use of nuclear power for energy.

Chapter 214 Thought Question 3 A 95% confidence interval for the proportion of adults in the U.S. who have diabetes extends from.07 to.11, or 7% to 11%. What does it mean to say that the interval from.07 to.11 represents a 95% confidence interval for the proportion of adults in the U.S. who have diabetes ?

Chapter 215 Thought Question 4 Would a 99% confidence interval for the proportion described in Question 3 be wider or narrower than the 95% interval given? Explain. (Hint: what is the difference between a 68% interval and a 95% interval?)

Chapter 216 Thought Question 5 In a May 2006 Zogby America poll of 1000 adults, 70% said that past efforts to enforce immigration laws have been inadequate. Based on this poll, a 95% confidence interval for the proportion in the population who feel this way is about 67% to 73%. If this poll had been based on 5000 adults instead, would the 95% confidence interval be wider or narrower than the interval given? Explain.

Chapter 217 Recall from previous chapters: Parameter fixed, unknown number that describes the population Statistic known value calculated from a sample a statistic is used to estimate a parameter Sampling Variability different samples from the same population may yield different values of the sample statistic estimates from samples will be closer to the true values in the population if the samples are larger

Chapter 218 Recall from previous chapters: Sampling Distribution tells what values a statistic takes and how often it takes those values in repeated sampling. Example: sample proportions ( ’s) from repeated sampling would have a normal distribution with a certain mean and standard deviation. Example : The amount by which the proportion obtained from the sample ( ) will differ from the true population proportion (p) rarely exceeds the margin of error.

Chapter 219 Case Study Science News, Jan. 27, 1995, p Comparing Fingerprint Patterns

Chapter 2110 Case Study: Fingerprints u Fingerprints are a “sexually dimorphic trait…which means they are among traits that may be influenced by prenatal hormones.” u It is known… –Most people have more ridges in the fingerprints of the right hand. (People with more ridges in the left hand have “leftward asymmetry.”) –Women are more likely than men to have leftward asymmetry. u Compare fingerprint patterns of heterosexual and homosexual men.

Chapter 2111 u 66 homosexual men were studied. 20 (30%) of the homosexual men showed left asymmetry. u 186 heterosexual men were also studied 26 (14%) of the heterosexual men showed left asymmetry. Case Study: Fingerprints Study Results

Chapter 2112 Case Study: Fingerprints A Question Assume that the proportion of all men who have leftward asymmetry is 15%. Is it unusual to observe a sample of 66 men with a sample proportion ( ) of 30% if the true population proportion (p) is 15%?

Chapter 2113 Twenty Simulated Samples (n=66) Sample Size Observed Proportion

Chapter 2114 The Rule for Sample Proportions If numerous simple random samples of size n are taken from the same population, the sample proportions from the various samples will have an approximately normal distribution. The mean of the sample proportions will be p (the true population proportion). The standard deviation will be:

Chapter 2115 Rule Conditions and Illustration u For rule to be valid, must have u Random sample u ‘Large’ sample size

Chapter 2116 Case Study: Fingerprints Sampling Distribution

Chapter 2117 Case Study: Fingerprints Answer to Question u Where should about 95% of the sample proportions lie?  mean plus or minus two standard deviations 0.15  2(0.044) = (0.044) =  95% should fall between & 0.238

Chapter Simulated Samples (n=66)

Chapter Simulated Samples (n=66) approximately 95% of sample proportions fall in this interval (0.062 to 0.238). Is it likely we would observe a sample proportion  0.30?

Chapter Simulated Samples (n=30)

Chapter Simulated Samples (n=30) approximately 95% of sample proportions fall in this interval. Is it likely we would observe a sample proportion  0.30?

Chapter 2122 Confidence Interval for a Population Proportion u An interval of values, computed from sample data, that is almost sure to cover the true population proportion. u “We are ‘highly confident’ that the true population proportion is contained in the calculated interval.” u Statistically (for a 95% C.I.): in repeated samples, 95% of the calculated confidence intervals should contain the true proportion.

Chapter 2123 u since we do not know the population proportion p (needed to calculate the standard deviation) we will use the sample proportion in its place. Formula for a 95% Confidence Interval for the Population Proportion (Empirical Rule) u sample proportion plus or minus two standard deviations of the sample proportion:

Chapter 2124 standard error (estimated standard deviation of ) Formula for a 95% Confidence Interval for the Population Proportion (Empirical Rule)

Chapter 2125 Margin of Error (plus or minus part of C.I.)

Chapter 2126 Formula for a C-level (%) Confidence Interval for the Population Proportion where z* is the critical value of the standard normal distribution for confidence level C

Chapter 2127 Common Values of z*

Chapter 2128 Case Study Brown, C. S., (1994) “To spank or not to spank.” USA Weekend, April 22-24, pp Parental Discipline What are parents’ attitudes and practices on discipline?

Chapter 2129 Case Study: Survey Parental Discipline u Nationwide random telephone survey of 1,250 adults. –474 respondents had children under 18 living at home –results on behavior based on the smaller sample u reported margin of error –3% for the full sample –5% for the smaller sample

Chapter 2130 Case Study: Results Parental Discipline u “The 1994 survey marks the first time a majority of parents reported not having physically disciplined their children in the previous year. Figures over the past six years show a steady decline in physical punishment, from a peak of 64 percent in 1988” –The 1994 proportion who did not spank or hit was 51% !

Chapter 2131 Case Study: Results Parental Discipline u Disciplining methods over the past year: –denied privileges: 79% –confined child to his/her room: 59% –spanked or hit: 49% –insulted or swore at child: 45% u Margin of error: 5% –Which of the above appear to show a true value different from 50%?

Chapter 2132 Case Study: Confidence Intervals Parental Discipline u denied privileges: 79% – : 0.79 –standard error of : –95% C.I.:.79  2(.019) : (.752,.828) u confined child to his/her room : 59% – : 0.59 –standard error of : –95% C.I.:.59  2(.023) : (.544,.636)

Chapter 2133 Case Study: Confidence Intervals Parental Discipline u spanked or hit: 49% – : 0.49 –standard error of : –95% C.I.:.49  2(.023) : (.444,.536) u insulted or swore at child: 45% – : 0.45 –standard error of : –95% C.I.:.45  2(.023) : (.404,.496)

Chapter 2134 Case Study: Results Parental Discipline u Asked of the full sample (n=1,250): “How often do you think repeated yelling or swearing at a child leads to long-term emotional problems?” –very often or often: 74% –sometimes: 17% –hardly ever or never: 7% –no response: 2% u Margin of error: 3%

Chapter 2135 Case Study: Confidence Intervals Parental Discipline u hardly ever or never: 7% – : 0.07 –standard error of : –95% C.I.:.07  2(.007) : (.056,.084) u Few people believe such behavior is harmless, but almost half (45%) of parents engaged in it!

Chapter 2136 Key Concepts (1 st half of Ch. 21) u Different samples (of the same size) will generally give different results. u We can specify what these results look like in the aggregate. u Rule for Sample Proportions u Compute and interpret Confidence Intervals for population proportions based on sample proportions

Chapter 2137 Inference for Population Means Sampling Distribution, Confidence Intervals u The remainder of this chapter discusses the situation when interest is in making conclusions about population means rather than population proportions –includes the rule for the sampling distribution of sample means ( ) –includes confidence intervals for one mean or a difference in two means

Chapter 2138 Thought Question 6 ( from Seeing Through Statistics, 2nd Edition, by Jessica M. Utts, p. 316) Suppose the mean weight of all women at a university is 135 pounds, with a standard deviation of 10 pounds. Recalling the material from Chapter 13 about bell-shaped curves, in what range would you expect 95% of the women’s weights to fall? 115 to 155 pounds

Chapter 2139 Thought Question 6 (cont.) If you were to randomly sample 10 women at the university, how close do you think their average weight would be to 135 pounds? If you randomly sample 1000 women, would you expect the average to be closer to 135 pounds than it would be for the sample of 10 women?

Chapter 2140 Thought Question 7 A study compared the serum HDL cholesterol levels in people with low-fat diets to people with diets high in fat intake. From the study, a 95% confidence interval for the mean HDL cholesterol for the low-fat group extends from 43.5 to a. Does this mean that 95% of all people with low-fat diets will have HDL cholesterol levels between 43.5 and 50.5? Explain.

Chapter 2141 Thought Question 7 (cont.) … a 95% confidence interval for the mean HDL cholesterol for the low-fat group extends from 43.5 to A 95% confidence interval for the mean HDL cholesterol for the high-fat group extends from 54.5 to b. Based on these results, would you conclude that people with low-fat diets have lower HDL cholesterol levels, on average, than people with high-fat diets? ( )

Chapter 2142 Thought Question 8 The first confidence interval in Question 7 was based on results from 50 people. The confidence interval spans a range of 7 units. If the results had been based on a much larger sample, would the confidence interval for the mean cholesterol level have been wider, more narrow or about the same? Explain.

Chapter 2143 Thought Question 9 In Question 7, we compared average HDL cholesterol levels for two diet groups by computing separate confidence intervals for the two means. Is there a more direct value (and single C.I.) to examine in order to make the comparison between the two groups?

Chapter 2144 Case Study Weights of Females at a Large University Suppose the mean weight of all women is  =135 pounds with a standard deviation of  =10 pounds and the weight values follow a bell- shaped curve. Hypothetical (from Seeing Through Statistics, 2nd Edition, by Jessica M. Utts, p. 316)

Chapter 2145 u What about the mean (average) of a sample of n women? What values would be expected? Case Study: Weights Questions u Where should 95% of all women’s weights fall?  mean plus or minus two standard deviations 135  2(10) = (10) = 155  95% should fall between 115 & 155

Chapter 2146 Twenty Simulated Samples (n=1000) Sample Size Observed Mean Weight

Chapter 2147 The Rule for Sample Means If numerous simple random samples of size n are taken from the same population, the sample means from the various samples will have an approximately normal distribution. The mean of the sample means will be  (the population mean). The standard deviation will be: (  is the population s.d.)

Chapter 2148 Conditions for the Rule for Sample Means u Random sample u Population of measurements… –Follows a bell-shaped curve - or - –Not bell-shaped, but sample is ‘large’

Chapter 2149 Case Study: Weights Sampling Distribution (for n = 10)

Chapter 2150 u Where should 95% of the sample mean weights fall (from samples of size n=10) ?  mean plus or minus two standard deviations 135  2(3.16) = (3.16) =  95% should fall between & Case Study: Weights Answer to Question (for n = 10)

Chapter 2151 Sampling Distribution of Mean (n=10)

Chapter 2152 Case Study: Weights Sampling Distribution (for n = 25)

Chapter 2153 u Where should 95% of the sample mean weights fall (from samples of size n=25) ?  mean plus or minus two standard deviations 135  2(2) = (2) = 139  95% should fall between 131 & 139 Case Study: Weights Answer to Question (for n = 25)

Chapter 2154 Sampling Distribution of Mean (n=25)

Chapter 2155 Case Study: Weights Sampling Distribution (for n = 100)

Chapter 2156 u Where should 95% of the sample mean weights fall (from samples of size n=100) ?  mean plus or minus two standard deviations 135  2(1) = (1) = 137  95% should fall between 133 & 137 Case Study: Weights Answer to Question (for n = 100)

Chapter 2157 Sampling Distribution of Mean (n=100)

Chapter 2158 Case Study Hypothetical Exercise and Pulse Rates Is the mean resting pulse rate of adult subjects who regularly exercise different from the mean resting pulse rate of those who do not regularly exercise? Find Confidence Intervals for the means

Chapter 2159 Case Study: Results Exercise and Pulse Rates A random sample of n 1 =31 nonexercisers yielded a sample mean of =75 beats per minute (bpm) with a sample standard deviation of s 1 =9.0 bpm. A random sample of n 2 =29 exercisers yielded a sample mean of =66 bpm with a sample standard deviation of s 2 =8.6 bpm.

Chapter 2160 The Rule for Sample Means If numerous simple random samples of size n are taken from the same population, the sample means from the various samples will have an approximately normal distribution. The mean of the sample means will be  (the population mean). The standard deviation will be: We do not know the value of  !

Chapter 2161 Standard Error of the (Sample) Mean SEM = standard error of the mean (standard deviation from the sample) = divided by (square root of the sample size) =

Chapter 2162 Case Study: Results Exercise and Pulse Rates u Typical deviation of an individual pulse rate (for Exercisers) is s = 8.6 u Typical deviation of a mean pulse rate (for Exercisers) is = 1.6

Chapter 2163 Case Study: Confidence Intervals Exercise and Pulse Rates u Nonexercisers :75 ± 2(1.6) = 75 ± 3.2 = (71.8, 78.2)  Exercisers: 66 ± 2(1.6) = 66  3.2 = (62.8, 69.2) u Do you think the population means are different? u 95% C.I. for the population mean:  sample mean  2  (standard error)  2  Yes, because the intervals do not overlap

Chapter 2164 Formula for a C-level (%) Confidence Interval for the Population Mean where z* is the critical value of the standard normal distribution for confidence level C

Chapter 2165 Careful Interpretation of a Confidence Interval u “We are 95% confident that the mean resting pulse rate for the population of all exercisers is between 62.8 and 69.2 bpm.” (We feel that plausible values for the population of exercisers’ mean resting pulse rate are between 62.8 and 69.2.) u ** This does not mean that 95% of all people who exercise regularly will have resting pulse rates between 62.8 and 69.2 bpm. ** u Statistically: 95% of all samples of size 29 from the population of exercisers should yield a sample mean within two standard errors of the population mean; i.e., in repeated samples, 95% of the C.I.s should contain the true population mean.

Chapter 2166 Exercise and Pulse Rates u 95% C.I. for the difference in population means (nonexercisers minus exercisers): (difference in sample means)  2  (SE of the difference) u Difference in sample means: = 9 u SE of the difference = 2.26 (given) u 95% confidence interval: (4.48, 13.52) – interval does not include zero (  means are different) Case Study: Confidence Intervals

Chapter 2167 An Experiment Testing a Vaccine for Those with Genital Herpes Case Study Adler, T., (1994) “Therapeutic vaccine fights herpes.” Science News, Vol. 145, June 18, p Does a new vaccine prevent the outbreak of herpes in people already infected?

Chapter 2168 An Experiment Testing a Vaccine for Those with Genital Herpes Case Study: Sample u 98 men and women aged 18 to 55 u Experience between 4 and 14 outbreaks per year u Experiment –Double-blind experiment –Randomized to vaccine or placebo

Chapter 2169 An Experiment Testing a Vaccine for Those with Genital Herpes Case Study: Report “The vaccine was well tolerated. gD2 recipients reported fewer recurrences per month than placebo recipients (mean 0.42 [sem 0.05] vs 0.55 [0.05]…)…”

Chapter 2170 An Experiment Testing a Vaccine for Those with Genital Herpes Case Study: Confidence Intervals u 95% C.I. for population mean recurrences: –Vaccine group: 0.42  2(0.05) : (.32,.52) –Placebo group: 0.55  2(0.05) : (.45,.65) u 95% C.I. for the difference in population means: –Difference = -0.13, SE = 0.07 (given) –C.I.: (-0.27, 0.01) (contains 0  means not different)

Chapter 2171 Key Concepts (2 nd half of Ch. 21) u Rule for Sample Means u Compute confidence intervals for means based on one sample u Compute confidence intervals for means based on two samples u Interpret Confidence Intervals for Means