Chapter 6 The Normal Distribution Section 6 1

Chapter 6 The Normal Distribution

Section 6 -1 Normal Distributions

Learning Target • IWBAT identify distributions as symmetric or skewed. • IWBAT identify the properties of a normal distribution.

Symmetric or Skewed • Symmetric – when the values are evenly distributed about the mean (mean, median, and mode are centrally located) • Negatively (left-skewed) – majority of the data values fall to the right of the mean (mean, median, and mode are located in this order) • Positively (right-skewed) – majority of the data values fall to the left of the mean (mode, median, and mean are located in this order)

Normal Distribution A normal distribution is a continuous, symmetric bell -shaped distribution of a variable. Properties 1. Bell-shaped 2. Mean, median, and mode are equal and located at the center 3. Unimodal 4. Symmetrical 5. Continuous 6. Curve never touches the x-axis 7. Total area under the curve is equal to 1 or 100% 8. Area that lies within 1 st. dev. from the mean is approx. . 68 or 68%, 2 st. dev. is approx. . 95 or 95%, and 3 st. dev. is approx. . 997 or 99. 7%

Standard Normal Distribution • A normal distribution that has a mean of 0 and a standard deviation of 1. • Uses z-scores to determine the number of deviations a value is from the mean. • Formula for z-score » Z= (value – mean)/ standard deviation • Use Table E (handout) – the chart ranges from -3. 49 to 3. 49 and gives area to four decimal places

How to find the area under the curve Step 1: Draw the normal distribution curve and shade the area. Step 2: Find the appropriate figure in the Procedure Table (next slide) and follow the directions

Examples • Find the area to the left of z = 1. 99 • Find the area to the right of z = -1. 16 • Find the area between z = 1. 68 and z = -1. 37 • Solutions – 0. 9767 – 1. 0000 – 0. 1230 = 0. 8770 – 0. 9535 – 0. 0853 = 0. 8682

Examples using Probability • Find the probability of each – P(0 < z < 2. 32) – P(z < 1. 65) – P(z > 1. 91) – – Solutions 0. 9898 – 0. 5000 = 0. 4898 or 48. 98% 0. 9505 or 95. 05% 1. 0000 – 0. 9719 = 0. 0281 or 2. 81%

Examples (finding z-score given area) • Find the z value such that the area under the standard normal distribution curve between 0 and the z value is 0. 2123. • Solution: add. 5000 to 0. 2123 because of the area to the left of 0. Then find the value in the chart and locate the z value that corresponds with 0. 7123, which is 0. 56.

Exercises 6 -1 1 – 47 odd

Section 6 -2 Applications of the Normal Distribution

Learning Target • IWBAT find probabilities for a normally distributed variable by transforming it into a standard normal variable.

Steps to Application Problems •

Example 1 •

Example 2 Each month a household generates of average of about 28 pounds of newspaper for garbage or recycling. Assume the standard deviation is 2 pounds. If a household is selected at random, find the probability of it generating a. Between 27 and 31 pounds per month b. More than 30. 2 pounds

Solution to Example 2 •

Example 3 (more than one is selected) The American Automobile Association reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is normally distributed and the standard deviation is 4. 5 minutes. If 80 calls are randomly selected, approximately how many will be responded to in less than 15 minutes?

Solution to Example 3 •

Find Data Values Given Specific Probabilities •

Example To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed.

Solution •

Exercises 6 -2 Problems 1 – 12 Then Problems 14 – 38 even