Understanding Sample Proportions and Confidence Intervals: A Gallup Poll Example, Study notes of Statistics

Download Understanding Sample Proportions and Confidence Intervals: A Gallup Poll Example and more Study notes Statistics in PDF only on Docsity! 1 Apr. 8 Statistic for the day: In the United States in 2001... Units of blood donated: 15 million Number of donors: 8 million Units transfused: 14 million Number of patients: 4.9 million Assignment: Prepare for midterm #3 Speaking of college basketball... For each of the following, say whether you are a fan of that sport or not: College basketball 41% answered yes or somewhat. The fine print (from gallup.com): + 3% margin of error; sample size=1003 In a Gallup poll conducted Dec. 5-8, 2004, people were asked: Formula for estimating the standard deviation of a sample proportion (don’t need histogram): sample proportion (1 sample proportion ) sample size × − .41 (1 .41) .016 1003 × − = If we happen to know the true population proportion we use it instead of the sample proportion. What to expect from sample proportions Facts: fingerprints may be influenced by prenatal hormones. Most people have more ridges on right hand than left. People who have more on the left hand are said to have leftward asymmetry. Women are more likely to have this trait than men. The proportion of all men who have this trait is about 15% In a study of 186 heterosexual and 66 homosexual men 26 (14%) heterosexual men showed the trait and 20 (30%) homosexual men showed the trait (Reference: Hall, J. A. Y. and Kimura, D. "Dermatoglyphic Asymmetry and Sexual Orientation in Men", Behavioral Neuroscience, Vol. 108, No. 6, 1203-1206, Dec 94. ) Is it unusual to observe a sample of 66 men and observe a sample proportion of 30%? We now know what the distribution of sample proportions based on a sample of 66 should look like. We will suppose that the true proportion in the population of men is 15%. 044 66 15115 .).(. =−×Standard deviation Thus, a sample proportion of 30% would be (.30-.15)/.044 = 3.41 standard deviations above the true mean, assuming that the sample is a representative sample from the population. 2 0.0 0.1 0.2 0.3 0 5 10 15 Fr eq ue nc y Histogram of proportions, with Normal Curve n = 66, true proportion = .15, standard deviation = .044 homosexual men 0.150.062 0.238 4 standard deviations 2 std devs The sample proportion for homosexual men (30%) is too large to come from the expected distribution of sample proportions. Sample means: measurement variables Data from stat 100 survey, spring 2004. Sample size 237. Mean value is 152.5 pounds. Standard deviation is about (240 – 100)/4 = 35 Suppose we want to estimate the mean weight at PSU 300200100 40 30 20 10 0 Weight Fr eq ue nc y Histogram of Weight, with Normal Curve Standard deviation is about (157 – 148)/4 = 9/4 = 2.25 160155150145 100 50 0 Weight Fr eq ue nc y curve, based on samples of size 237 Histogram of 1000 means with normal Hypothetical result, using a “population” that resembles our sample: Extremely interesting: The histogram of means is bell-shaped, even though the original population was skewed! Formula for estimating the standard deviation of the sample mean (don’t need histogram) Just like in the case of proportions, we would like to have a simple formula to find the standard deviation of the mean without having to resample a lot of times. Suppose we have the standard deviation of the original sample. Then the standard deviation of the sample mean is: standard deviation of the data sample size Example: SAT math scores Suppose nationally we know that the SAT math test has a mean of 100 points and a standard deviation of 100 points. Draw by hand a picture of what you expect the distribution of sample means based on samples of size 100 to look like. Sample means have a normal distribution mean 500 standard deviation 100/10 = 10 So draw a bell shaped curve, centered at 500, with 95% of the bell between 500 – 20 = 480 and 500 + 20 = 520 A sample of 100 SAT math scores with a mean of 540 would be very unusual. A sample of 100 with a mean of 510 would not be unusual. 460 480 500 520 540 0. 00 0. 01 0. 02 0. 03 0. 04 Normal curve of SAT means, sample size 100 Score 5 Example: Estimate mean # of pairs of jeans owned by a student at PSU Histogram of Jeans Jeans Fr eq ue nc y 0 10 20 30 40 0 10 20 30 40 50 Mean = 7.8 pairs St. Dev. = 5.8 pairs Sample size = 222 Give a 98% confidence interval. Example: Estimate mean # of pairs of jeans owned by a student at PSU Mean = 7.8 pairs St. Dev. = 5.8 pairs Sample size = 222 Give a 98% confidence interval. 5.8SEM 0.4 222 = = # of SEMs for 98% confidence: 2.33 98% confidence interval: 7.8 ± 2.33×0.4 7.8 ± 0.9, or 6.9 to 8.7 Interpretation: We estimate that the population of Penn State students owns 7.8 pairs of jeans on average. 98% confidence interval is 6.9 to 8.7 pairs, a reasonable range of values for the true (population) mean. Guess the next numbers in the sequence 1, 1, 2, 3, 5, 8, 13, Called a Fibonacci sequence. Ratios of pairs after a while equal approximately .618 eg. 8/13 = .615 13/21 = .619 21/34 = .618 Fibonacci Sequence 21, 34, ... width length 618.= length widthIf then the rectangle is called a golden rectangle. Width to Length ratios for rectangles appearing on beaded baskets of the Shoshoni 0.693 0.662 0.690 0.606 0.570 0.749 0.652 0.628 0.609 0.844 0.654 0.615 0.668 0.601 0.576 0.670 0.606 0.611 0.553 0.633 0.625 0.610 0.600 0.633 0.595 0.85 0.75 0.65 0.55 C 1 beaded baskets Width to Length ratio of rectangles in Shoshoni Golden Rectangle: .618 6 Question: Is the golden rectangle (.618) a reasonable value for the mean of the population of Shoshoni rectangles? 1. sample mean: .638 2. sample standard deviation: SD = .061 3. sample size: 25 4. standard error of the mean: SEM = .012 (I calculated it for you.) 95% confidence interval for the true mean: .638 ± 2×.012, which is .614 to .662. Back to holding babies on the left. Accepting that Lee Salk has presented a strong case for holding babies on the left, what is the selective advantage from the point of view of evolution? Hypothesis: Holding baby on the left is holding baby over the heart. And the sound of a human heartbeat is soothing to baby. Babies were divided into three groups: •light birth weight (2510 – 3000 grams), •medium birth weight (3010 – 3500 grams) and •heavy (3510 grams and above). The weight change from day 1 to day 4 was recorded. We want to know if the population means for treatment (heartbeat) and control (no heartbeat) are close or not. To test this hypothesis, Salk randomly selected a period of 4 days and played the sound of a heart beating in a new baby nursery. Then he did the same without the heartbeat for a new group of newborns. Control SEM: 11.33 Treatment SEM: 8.45 2 28.45 11.33 14.13+ = Light birth weight 2860-20Control (no heartbeat) 355065Treatment (heartbeat) samp. sizeStd. Dev.Mean 11.33 8.45 SEM Light birth weight Difference in sample means: 85 g Standard error of the difference: 14.13 g 95% confidence interval for the true difference: 85 ± 2×14.13, which is 56.7 to 113.26 Since zero is not in the CI, zero is not a plausible value. Thus, we have found a statistically significant difference. The researchers found: •musicians with perfect pitch: mean dPT = -.57 •musicians without perfect pitch: mean dPT = -.23 Question: Are the dPT means close or not? Is there a difference between musicians with and without perfect pitch? Equivalently we ask: Is the difference in means: -.57 – (-.23) = -.34 close to 0? Perfect pitch and the brain 7 SE of diff SEM SD sample size means .039.019 .17.21 1911 -.23-.57 musicians no perf pitch musicians perf pitch Diff in means = -.57 – (-.23) = -.34 95% CI: -.34 ± 2×.043, which is -.43 to -.26 Conclusion: They are not close. There is a difference. 2 2.019 .039 .043+ = General conclusions: There is a significant difference between the asymmetry of the PT for musicians with perfect pitch and both musicians without perfect pitch and non-musicians. This strongly suggests that there is a relationship between the physical structure of the PT in the brain and perfect pitch ability. Categorical ordinal, categorical nominal, quantitative discrete, or quantitative continuous? Eye color Weight Number of siblings Gender Time in 100-meter dash Number of cigarettes smoked in a day Building where your first class occurs Year in school (fr / so / jr / sr) (CN) (QC) (QD) (CN) (QC) (QD) (CN) (CO) Consider a clock that’s 5 minutes fast. Valid or invalid? Reliable or unreliable? Biased or unbiased? Answer: valid, reliable and biased. Consider a scale that is sometimes several pounds too low, sometimes several pounds too high Valid or invalid? Reliable or unreliable? Biased or unbiased? Answer: valid, unreliable and unbiased.