4/19/2009. Probability Distributions. Inference. Example 1. Example 2. Parameter versus statistic. Normal Probability Distribution N

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Probability Distributions Normal Probability Distribution N Chapter 6 Inference It was reported that the 2008 Super Bowl was watched by 97.5 million people. But how does anyone know that?

The answer is that a survey was taken, and the results were used to project a total number of viewers for the entire country.

Parameter versus statistic Population: the entire group of individuals in which we are interested but can t usually assess directly.

A statistic is a number describing a characteristic of a sample. We often use a statistic to estimate an unknown population parameter.

They found no elevated levels of any harmful substances.

1 Probability Distributions Normal Probability Distribution N Chapter 6 Inference It was reported that the 2008 Super Bowl was watched by 97.5 million people. But how does anyone know that? They certainly didn t ask everyone in the United States whether they watched it or not! The answer is that a survey was taken, and the results were used to project a total number of viewers for the entire country. This is an eample of statistical inference using the results of a sample to determine something about the entire population. Parameter versus statistic Population: the entire group of individuals in which we are interested but can t usually assess directly. A parameter is a number describing a characteristic of the population. Parameters are usually unknown. Sample: the part of the population we actually eamine and for which we do have data. A statistic is a number describing a characteristic of a sample. We often use a statistic to estimate an unknown population parameter. 3 4 Eample 1 The Environmental Protection Agency took soil samples at 20 locations near a former industrial waste dump and checked each for evidence of toic chemicals. They found no elevated levels of any harmful substances. Population: ALL the soil near the waste dump Sample: the 20 soil samples Parameter: mean level of toic chemicals in the ground around the waste dump Statistic: the mean level of toic chemicals in the 20 soil samples Eample 2 In the days and weeks before the 2008 Presidential Election, many organizations made predictions that Barack Obama would win the election over John McCain by a certain number of percentage points. How did they do it? Again, statistical inference: they took polls of a sample of potential voters, and used the results of their polls to predict what would happen on Election Day. Population: all American voters Sample: the sample of American voters polled Parameter: proportion of all American voters who would vote for Obama Statistic: proportion of individuals in the poll who would vote for Obama 5 6 1

Accuracy of an estimate The accuracy of an estimate will depend on the sampling method. The sample must be chosen randomly. The accuracy of an estimate will depend on the size of the sample.

Like for a cook: it doesn t matter how big the pot of soup is, to taste it, he only needs a spoonful.

2 Accuracy of an estimate The accuracy of an estimate will depend on the sampling method. The sample must be chosen randomly. The accuracy of an estimate will depend on the size of the sample. BUT the size of the population has almost nothing to do with the accuracy of a statistic that is estimating a population parameter. Like for a cook: it doesn t matter how big the pot of soup is, to taste it, he only needs a spoonful. Notation Variable of interest: Categorical Then we are interested in PROPORTION Notation: Population parameter: p Sample statistic : $p Variable of interest: Quantitative Then we are interested in MEAN Notation: Population parameter: µ Sample statistic: 7 8 Sampling Variability When we take many samples, the statistics from the samples are usually different from the population figures, and also different from what we got in the first sample. This very intuitive idea, that sample results change from sample to sample, is called sampling variability. Comments 1. Parameters are usually unknown, because it is impractical or impossible to know eactly what values a variable takes for every member of the population. 2. Statistics are computed from the sample, and vary from sample to sample due to sampling variability Sampling Distributions The sampling distribution is a distribution of a sample statistic in infinite number of samples. Sampling distribution of the sample mean, Sampling distribution of Histogram of some sample averages e.html

3 OK, we have the sampling distribution of the sample means. Then what? Sampling distributions, like data distributions, are best described by shape, center, and spread. Shape, Center, and Spread Shape: Many, but not all, sampling distributions are approimately normal. Center: The mean will be denoted by µ with a subscript to indicate which sampling distribution is being discussed. For eample, the mean of the sampling distribution of the mean is represented by the symbol µ X. (The mean of the sample means.) Spread: the standard deviation of the σ X sampling distribution of the sample means and is Mean and standard error of the sampling distribution of the sample means Suppose that is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the sampling distribution of has mean and standard deviation µ = µ σ σ = n For any population with mean µ and standard deviation σ: The mean, or center of the sampling distribution of, is equal to the population mean µ. The standard deviation of the sampling distribution is σ/ n, where n is the sample size. Sampling distribution of σ/ n µ Mean of a sampling distribution of There is no tendency for a sample mean to fall systematically above or below µ, even if the distribution of the raw data is skewed. Thus, the mean of the sampling distribution of is an unbiased estimator of the population mean μ it will be correct on average in many samples. Standard error of a sampling distribution of The standard deviation of the sampling distribution measures how much the sample statistic varies from sample to sample. It is smaller than the standard deviation of the population by a factor of n. Averages are less variable than individual observations

Generating Sampling Distributions 1. Take a random sample of a fied size n from a population. 2. Compute the summary statistics (mean, proportion). 3. Repeat steps 1 and 2 many times. 4.

4 Generating Sampling Distributions 1. Take a random sample of a fied size n from a population. 2. Compute the summary statistics (mean, proportion). 3. Repeat steps 1 and 2 many times. 4. Display the distribution of the summary statistics. Eample Etensive studies have found that the DMS odor threshold of adults follows a roughly normal distribution with mean µ =25 micrograms per liter and standard deviation σ =7 micrograms per liter. With this information, we can simulate many runs of our study with different subjects drawn at random from the population. We take 1000 samples of size 10, find the 1000 sample mean thresholds, and make a histogram of these 1000 values The results from the 1000 samples 1 st SRS of size 10: 2 nd SRS of size 10: 3 rd SRS of size 10: M = 36, s = 32. = 22. 8, s = 2. 7 = 30. 4, s = 41. The sampling distribution of the statistic. Frequency C Shape: looks normal. Center: the mean of the 1000 s is µ = The distribution is centered very close to the population mean µ = th SRS of size 10: = 28. 9, s = 21. Spread: the standard error of the 1000 smaller than the standard deviation σ = 7 s is 2.191, notably of the population For normally distributed populations When a variable in a population is normally distributed, then the sampling distribution of for all possible samples of size n is also normally distributed. If the population is N(µ,σ), then the sample means distribution is N(µ,σ/ n ). 23 Sample means Population 24 IQ scores: population vs. sample In a large population of adults, the mean IQ is 112 with standard deviation 16. Suppose 100 adults are randomly selected for a market research campaign. The distribution of the sample mean IQ is A) eactly normal, mean 112, standard deviation 16. B) approimately normal, mean 112, standard deviation 16. C) approimately normal, mean 112, standard deviation 1.6. D) approimately normal, mean 112, standard deviation 4. C) approimately normal, mean 112, standard deviation 1.6. Population distribution: N (µ = 112; σ = 16) Sampling distribution for n = 200 is N (µ = 112; σ / n = 1.6) μ σ n 4

5 Application Hypokalemia is diagnosed when blood potassium levels are low, below 3.5mEq/dl. Let s assume that we know a patient whose measured potassium levels vary daily according to a normal distribution N(µ = 3.8, σ = 0.2). If only one measurement is made, what's the probability that this patient will be misdiagnosed hypokalemic? ( µ ) = = σ 0.2 z z = 1.5, P(z < 1.5) = % If instead measurements are taken on four separate days, what is the probability of such a misdiagnosis? Note: ( µ ) z = = σ n z = 3, P(z < 1.5) = % Make sure to standardize (z) using the standard deviation for the sampling distribution But Not all variables are normally distributed. Income is typically strongly skewed for eample. Is still a good estimator of µ then? The Central Limit Theorem will rescue us! The Central Limit Theorem VERY IMPORTANT!!! When randomly sampling from any population with mean µ and standard deviation σ, when n is large enough, the sampling distribution of approimately normal: N(µ, σ/ n). is Central Limit Theorem The Central Limit Theorem guarantees that a distribution of sample mean to be approimately normal as long as the sample size is large enough. We will depend on the Central Limit Theorem again and again in order to take advantage of normal probability calculations when we use sample mean to draw conclusions about population mean, even if the population distribution is not normal Comments There is no requirement on the shape of the population distribution. This is where the strength of the Central Limit Theorem lies. It tells us that regardless of the shape of the population distribution, averages that are based on a large enough sample will have a normal distribution

The central limit theorem Population with strongly skewed distribution Sampling distribution of for n = 10 observations Sampling distribution of for n = 2 observations Sampling distribution of for n

6 The central limit theorem Population with strongly skewed distribution Sampling distribution of for n = 10 observations Sampling distribution of for n = 2 observations Sampling distribution of for n = 25 observations Assessing Normality A normal probability plot is a graph with the original set of data on the -ais, and the corresponding z scores for each data value on the y- ais. If the points appear to lie reasonably close to a straight line and there does not appear to be a systematic pattern that is not a straight line, we can conclude that the data came from a normally distributed population Normal distribution v right-skewed distribution left-skewed distribution Short-tailed distribution Long-tailed distribution 33 6

3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability

3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability Random variable The outcome of each procedure is determined by chance. Probability Distributions Normal Probability Distribution N Chapter 6 Discrete Random variables takes on a countable number of values