Introduction to Biostatistics
Introduction to Biostatistics STAT 541
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Haley J Schuhl
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Mrs. Triston Collier
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This 6 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 541 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 48 views. For similar materials see /class/205085/stat-541-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.
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Date Created: 09/17/15
Continuous Probability Distributions Recall that we have contrasted continuous variables from discrete variab es by noting that while there are gaps between possible values of a discrete variable7 a continuous variable can take any value in some interval of the number line Suppose X is a discrete random variable taking on the values 12n with probability distribution function Then Pl S X S 4 fXQ fX3 fX4 Suppose X is a continuous random variable with probability density function Then PaltXlt bfzdz 1 3 A random variable X is said to follow a Gaussian Normal distribution with population parameters 1 and 0 if its probability density function is given by 1 7 I 7 W W m ET A random variable X is said to follow the standard Normal distribution if its population parameters 1 and 0 equal 0 and 17 respectively ln this case7 1 712 361 9 2 Suppose X is a continuous random variable taking on any value x E 6 Then the probability density function satis es fXz gt O ifzecd 0 otherwise 1 A fxudu1 The probability Pa lt X lt 1 equals fXltugtdu Ebgample Normal distribution Suppose X represents blood pressure and suppose that the population mean u 129 and the population standard deviation 0 198 PX gt150 Ag 927702 oo 1 7 z 7 1292 27r1982 6 2698 The probability that a Normal random variable takes on a value within an interval is equal to the area under the part of the normal density which lies above the interval Unfortunately there is no simple formula for calculating this area so we need to use a table Fortunately we only need a table for the standard normal density with mean u O and standard deviation 0 1 7 Two simple rules can be very helpful in calculating normal probabilities 0 Since the total area under any density is 1 PZgtz17PZgz 0 Since the normal density is symmetric about 0 PZ lt z PZ gt 72 Plz ill 6 If we want to calculate probabilities for a general normal random variable X with mean u and SD 0 we need to construct a new random variable called the standardized score of X H a Z Given values for u and a we can actually go back and forth between the X scale 7 and the Z scale 7 XMUZ 3 Suppose that X is a random variable that represents height For the population of 18 to 74 year old women height is normally distributed with mean u 689 inches and standard deviation 0 26 inches If we randomly select a woman from this population what s the probability that she is between 60 and 68 inches tal 7 9 Suppose serum cholesterol levels X for children in Wisconsin have mean 175mg100ml and SD SOmg100ml Suppose we want to know the limits within which 95 of the population lies We know PZ gt 196 0025 so that P7196 g Z 196 095 What kinds of questions can you now answer 7 Use BP as an illustration Suppose we know BP is normally distributed with a speci c mean u and variance 02 1 A person walks in and you record the BP You can tell if this person has normal 7 BP or is an outlier 2 You can tell what proportion of people lie inside or outside a given range 3 lf say 20 men come in to the of ce on a given day and each has his BP taken you can tell the probability that at least one at most 2 at least 5 etc lie outside or inside some given range 10 Among females in the United States between 18 and 74 years of age diastolic blood pressure is normally distributed with mean u 77 mmHG and standard deviation 0 116 mmHg 1 What is the probability that a randomly selected woman has a diastolic blood pressure less than 60 mmHg 2 What is the probability that a randomly selected woman has a diastolic blood pressure greater than 90 mmHg 3 What is the probability that among ve women selected at random from the population at least one will have a pressure outside the range 60 to 90 mmHg 7 To answer these questions we ve made some key assumptions We have known 7 that the populations of interest are Normally distributed We have known 7 the population mean u and the population standard deviation 0 Most of the time we don t know these things So most often we collect a random independent sample from a population and estimate population parameters of interest 14 Statistical Inference Q l happen to know 7 BP levels for all men in the US follows a normal distribution with mean u and standard deviation 0 The process of drawing conclusions about an entire population based on the information in a sample is known as statistical inference You need to guess at u and a How would you do this 7 16 Histograms of 100 sample means for different sample sizes Q l happen to know that the number of car accidents in Madison each year follows a Poisson distribution l know the mean and so l 0 if know the variance ll You need to guess at the mean How would you do this 7 O i i in 15 20 25 O l 15 25 1 l O 15 25 Histograms of 100 sample means for different sample sizes mm z O 15 20 25 15 Histograms of 100 sample means for different sample sizes 15 20 25 8 C JD 15 20 25 13 Q l happen to know that the number of successful surgeries out of 10000 follows a Binomial distribution I know the mean and variance You need to guess at the mean and variance How would you do this 7 2a The probability distribution of X is called the sampling distribution of X Understanding properties of the sampling distribution of X al ows us to make inference about population parameters based on a single sample Characteristics that we observed in histograms which approximate the sampling distribution of X 1 The mean of the sampling distribution is near the population mean from which the samples were taken sample size 72 doesn t matter 2 The variance ofthe sampling distribution gets smaller as the size of the sample n increases 3 For large sample sizes n the sampling distribution looks normal CENTRAL LlMlT THEOREM Let X1X2 XyL denote n independent random variables sampled from the same distribution which has a nite mean 1 and variance 02 If n is large7 then Ma N UN 2 No1 ln other words7 X is approximately Normally distributed with mean 02 1 and variance 7 Approximation gets better as 72 increases Recall the questions stated earlier on BP ls a given BP typical 7 What proportions of BPs lie in a given range 7 lf you see 20 men in a day7 what s the probability that at least one will have BP outside a given range 7 To answer these questions7 we needed to assume that BP is normally distributed with a speci c mean 1 and variance 02 Thanks to the CLT7 we can now answer similar questions without assuming a Normal distribution 22 NOTE 7 independent samples from the same distribution are often called independent and identically distributed iid 24 Q Consider the distribution of cholesterol levels for US men aged 20 7 74 Assume the mean 1 211 mg100ml and the standard deviation is U 46 mg100ml Select a sample of size n from the population ls the sample an outlier 7 Does it have an unusually high or low sample mean 7 To answer this7 we could gure out the interval that encloses say 95 of the sample means and see if the sample mean from our sample falls within that range So for a xed 72 we want to nd I and In such that Pz g Xn g mu 095 We know P7196 g Z 196 095 For n 25 P ltXlt P7196lt lt196 in w 1 4W 1 46 7 46 P7196 ltX7211lt196 l l 25 7 v25 P211719692 g X g 211 196 92 M19297 g X g 22903 X 7 211 25
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