Introductory Statistics Introductory Statistics

Outline of Lesson 05 Videos Density Curve 68-95-99.7 Rule Z-Score Converting Z-Score to Probability Percentiles for Normal Distribution QQ-Plots – Assessing Normality

The total area under the curve equals 1. Density Curve The total area under the curve equals 1. The density curve always lies on or above the horizontal axis. Area inside curve equals 1. Horizontal Axis

Outline of Lesson 05 Videos Density Curve 68-95-99.7 Rule Z-Score Converting Z-Score to Probability Percentiles for Normal Distribution QQ-Plots – Assessing Normality

68-95-99.7 Rule The empirical rule: Approx. 68% of all of the data fall within 1 standard deviation of the mean Approx. 95% of all of the data fall within 2 standard deviations of the mean Approx. 99.7% of all of the data fall within 3 standard deviations of the mean Mean - (# of Std Deviations * σ) = Lower Value Mean + (# of Std Deviations * σ) = Upper Value Sketching the normal distribution is strongly recommended. The mean incubation time of fertilized chicken eggs kept at a certain temperature is 21 day. Incubation times are normally distributed with a standard deviation (σ) of one day. According to the Empirical Rule, approximately 95% of incubations will have times between which two values? 21 +- 2 (1) = (19,23)

68-95-99.7 Rule

68-95-99.7 Rule The council for Graduate Medical Education found that medical residents’ mean number of hours worked in a week is 81.7 The number of hours worked per week is normally distributed with a standard deviation of 6.9 hours. According to the Empirical Rule, approximately 95% of students will have work times per week between which two values? 95% of Data will fall within: 81.7 +- 2*(6.9) = (67.9,95.5) What about 99.7% of students? 99.7% of Data will fall within: 81.7 +- 3*(6.9) = (61,102.4) What approximate percent of data will fall between 74.8 hours and 88.6 hours? Both 74.8 and 88.6 are 1 standard deviation from the mean so 68% of the data will fall within those two points

68-95-99.7 Rule

Outline of Lesson 05 Videos Density Curve 68-95-99.7 Rule Z-Score Converting Z-Score to Probability Percentiles for Normal Distribution QQ-Plots – Assessing Normality

Standard Normal Distribution Suppose the variable X is normally distributed with mean µ and standard deviation σ. Then the random variable Z is normally distributed with mean 0 and standard deviation of 1. The Z-Score is calculated as follows: Z-Score represents the number of standard deviations a value (X) is away from the mean (µ). Uses for the Z-Score Determine values that may be extreme Compare values from two different types of normal distributions Being capable of using normal probabilities to calculate the probabilities of specific events

Determining Extreme Values The mean incubation time of fertilized chicken eggs kept at a certain temperature is 21 day. Incubation times are normally distributed with a standard deviation of one day. What is the Z-Score of an incubation time of 20.5 days? Would that be considered extreme? How about 25 days? Z=20.5-21/1 = -0.5 – Not too extreme Z=25-21/1 = 4 Appears to be extreme The council for Graduate Medical Education found that medical residents’ mean number of hours worked in a week is 81.7 The number of hours worked per week is normally distributed with a standard deviation of 6.9 hours. What is the Z-Score of a study time in a week of 84 hours? Would that be considered extreme? How about 110 hours? Z=(84-81.7)/6.9 = 0.33 - Not too extreme Z=(110-81.7)/6.9 = 4.10 - Appears to be extreme

Comparing Scores across different distributions Who had the best baseball season?

Outline of Lesson 05 Videos Density Curve 68-95-99.7 Rule Z-Score Converting Z-Score to Probability Percentiles for Normal Distribution QQ-Plots – Assessing Normality

Steps to Finding Area or Probability Under Normal Curve Convert (standardize) the values of X using the standard normal variable Z (Find the Z-Score) Find the area under the standard normal curve by using the applet http://byuimath.com/apps/normprob.html

Steps to Finding Area or Probability Under Normal Curve The length of human pregnancies from conception to birth is normally distributed with mean 266 Days and a standard deviation of 16 days. What is the probability of a pregnancy lasts less than 240 day? Convert (standardize) the values of X using the standard normal variable Z (Find the Z-Score) Find the area under the standard normal curve (Using the applet) Z = (240 – 266) / 16 = -1.625 0.052

Steps to Finding Area or Probability Under Normal Curve The length of human pregnancies from conception to birth is normally distributed with mean 266 Days and a standard deviation of 16 days. What is the probability of a pregnancy lasts more than 270 days? Convert (standardize) the values of X using the standard normal variable Z (Find the Z-Score) Find the area under the standard normal curve (Using the applet) Z = (270 – 266) / 16 = 0.25 0.401

Steps to Finding Area or Probability Under Normal Curve The length of human pregnancies from conception to birth is normally distributed with mean 266 Days and a standard deviation of 16 days. What is the probability that a pregnancy lasts between 241 and 291 days? Convert (standardize) the values of X using the standard normal variable Z (Find the Z-Score) Find the area under the standard normal curve (Using the applet) Z = (241 – 266) / 16 = -1.563 Z = (291 – 266) / 16 = 1.563 0.882

Outline of Lesson 05 Videos Density Curve 68-95-99.7 Rule Z-Score Converting Z-Score to Probability Percentiles for Normal Distribution QQ-Plots – Assessing Normality

Steps to Finding a value when given the area or percentile (less than 50th Percentile) If needed, click on the areas with the normal distribution curve so that the only area shaded in blue is on the left side of the left line Type in the Percentile number in decimal form in the area box. (e.g. 40th Percentile) Use the Z score under the left red line and put into the formula X = µ + (Z*σ) to find the value of interest (X) which is the percentile you are looking for (e.g. µ = 266 Days and σ = 16 days for human pregnancies. X = µ + (Z*σ) = 266 + (-.253 * 16) = 261.95

Steps to Finding a value when given the area or percentile (greater than 50th Percentile) If needed, click on the areas with the normal distribution curve so that the areas shaded in blue are on the left side of the left line and in between both red lines Type in the Percentile number in decimal form in the area box. (e.g. 90th Percentile) Use the Z score under the right red line and put into the formula X = µ + (Z*σ) to find the value of interest (X) which is the percentile you are looking for (e.g. µ = 266 Days and σ = 16 days for human pregnancies. X = µ + (Z*σ) = 266 + (1.28 * 16) = 286.48

Outline of Lesson 05 Videos Density Curve 68-95-99.7 Rule Z-Score Converting Z-Score to Probability Percentiles for Normal Distribution QQ-Plots – Assessing Normality

QQ Plot Normally Distributed Not Normally Distributed