# Lecture 5 - Normal Distributions, Sampling Distribution of the Mean, & Z-Scores Print-out

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NORMAL DISTRIBUTIONS, SAMPLING DISTRIBUTION OF THE MEAN, & Z-SCORES John Pineda, MPH HSCI 390: Biostatistics 1 Objectives Know the properties of the normal distribution. Know how to calculate z-scores and determine the proportion of scores Greater than/equal to a given z-score Less than/equal to a given z-score Between two z-scores Understand the standard normal distribution table: looking up z-scores based on given probability, and probability based on the z-score. Understand the central limit theorem and how it applies to sample distributions. 2 Normal Distribution - Properties Symmetry : Same shape on both halves, if you folded the distribution in half lengthwise (mirror image). Unimodality : Only one mode, and it equals the mean and median. Shape : It has a very distinct bell curve, where specific parts of the bell curve always represent a fixed proportion of the entire distribution. This applies for continuous variables, not discrete! 3 Z-Score Distribution It's a version of the normal distribution (called the standard normal distribution) where the scores (answers) are changed such that the mean always = 0 & standard deviation always = 1 . Why use the z-distribution instead of std. deviation? It can tell you exactly how far away or how close a given score is to the mean. You can calculate probabilities & percentiles . 4
Z-Score Calculation A given score on a normal distribution is expressed as the number of standard deviation units away from the mean. Known as Z-scores, or standardized scores x = Individual Score ̅࠵? ࠵?࠵? ࠵? = Mean ࠵? ࠵?࠵? ࠵? = Standard Deviation ࠵? = !" ̅! \$ or ࠵? = !"% & **Doesn't matter which formula you use, however it will matter later for hypothesis testing!! 5 Example #1 A television cable company receives numerous phone calls throughout the day from customers reporting service troubles and from would-be subscribers to the cable network. Most of these callers are put "on hold" until a company operator is free to help them. The company has determined that the length of time a caller is on hold is normally distributed with a mean of 3.1 minutes and a standard deviation 0.9 minutes. Company experts have decided that if as many as 5% of the callers are put on hold for 4.8 minutes or longer, more operators should be hired. What percentage of the company's callers are put on hold for more than 4.8 minutes? Should the company hire more operators? Show these probabilities on a sketch of the normal curve. 6 Example #1 - Continued Given x = 4.8 minutes, ̅࠵? = 3.1 minutes, s = 0.9 minutes ** Write out the given information, and draw the curve!! ࠵? ࠵? > 4.8 = ࠵? ࠵? > !"! # = ࠵? ࠵? > \$.&"'.( ).* = ࠵? ࠵? > (.+ ).* = ࠵?(࠵? > 1.89) Since it is "greater," you subtract the probability you get from 1: 1- 0.9706 = 0.0294 [Percentage: 0.0294 x 100 = 2.94%] 7 Example #2 A public health researcher at St. Jude's Hospital wanted to know more about a patient's hypertension problems. Suppose a mild hypertensive is defined as a person whose diastolic blood pressure (DBP) is between 90 and 100 mm Hg inclusive, and the subjects are 35- to 44-year-old men whose blood pressures are normally distributed with mean 80 and variance 144. What is the probability that a randomly selected person from this population will be a mild hypertensive? 8
Example #2 - Continued ࠵? 90 < ࠵? < 100 = ࠵? ࠵? − ̅࠵? ࠵? < ࠵? < ࠵? − ̅࠵? ࠵? = ࠵? 90 − 80 12 < ࠵? < 100 − 80 12 = ࠵? 0.83 < ࠵? < 1.67 You find the probability that corresponds to both 0.83 and 1.67; 0.83 = 0.7967 and 1.67 = 0.9525 Find the range based on those: 0.9525 - 0.7967 = 0.1558 [percentage: 0.1558 x 100 = 15.58%] Given x 1 = 90 mm Hg, X 2 = 100 mm Hg, ࠵? = 80 mm Hg, s 2 = 144 mm Hg 9 Example #3 A treatment trial is proposed to test the efficacy of vitamin E as a preventive agent for cancer. One problem with such a study is how to assess compliance among participants. A small pilot study is undertaken to establish criteria for compliance with the proposed study agents. In this study 10 patients are given 400 IU/day of vitamin E and 10 patients are given similar-sized tablets of placebo over a 3-month period. Their serum vitamin-E levels are measured before and after the 3- month period and the change 3-month baseline is shown. What percentage of the vitamin E group would be expected to show a change of not more than 0.30 mg/dL? 10 Example #3 - Continued Given x = 0.30 mg/dL, ࠵? = 0.80 mg/dL, s = 0.48 mg/dL ࠵? ࠵? < 0.30 = ࠵? ࠵? < ࠵? − ̅࠵? ࠵? = ࠵? ࠵? < 0.30 − 0.80 0.48 = ࠵? ࠵? < "#.%# #.&' = ࠵?(࠵? < −1.04) Since it is "Less," the probability you find is the answer: 0.1492 [Percentage: 0.1492 x 100 = 14.92%] 11 Finding Original Score ࠵? = ࠵? + ࠵? ࠵? or ࠵? = ̅࠵? + ࠵? ࠵? x = Individual Score ̅࠵? ࠵?࠵? ࠵? = Mean ࠵? ࠵?࠵? ࠵? = Standard Deviation If you're given a percentage or proportion, convert that to a z-score. Look at the table! 12
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