Section 5.1: Probability and area
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1 Section 5.1: Probability and area
2 Review Normal Distribution s z = x - m s Standard Normal Distribution s=1 m x m=0 z The area that falls in the interval under the nonstandard normal curve is the same as that under the standard normal curve within the corresponding z-boundaries. The area under the standard normal curve to the left of a z-score gives the probability that z is less than that z- score.
3 Example: Finding Probabilities for Normal Distributions A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a standard deviation of 12 minutes in the store. The length of time spent in the store is normally distributed and is represented by the variable x. A shopper enters the store. Find the probability that the shopper will be in the store more than 39 minutes.
4 Solution: Finding Probabilities for Normal Distributions Normal Distribution μ = 45 σ = 12 P(x > 39) x m z = = = 05. s 12 Standard Normal Distribution μ = 0 σ = 1 P(z > 0.5) x z P(x > 39) = P(z > 0.5) =?
5 Example: Finding Probabilities for Normal Distributions If 200 shoppers enter the store, how many shoppers would you expect to be in the store more than 39 minutes? Solution: Recall P(x > 39) = (0.6915) =138.3 (or about 138) shoppers
6 Use technology to find Normal Probabilities TI-83/84 Plus: normalcdf(lower bound, upper bound, mean, standard deviation) If you want all the numbers greater than a certain value, your upper boundary will be positive infinity. Use a large positive number like If you want all the numbers less than a certain value, your lower boundary will be negative infinity. Use a large negative number like Do not ever use normalpdf! No need to computer z-scores using TI-83/84.
7 Example: Find Normal Probabilities IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the probability a randomly selected person has an IQ score between 100 and 120. Choose the closest answer. A B Use the table and the technology. Discuss with your classmates if you don t know how to get the answer. C D
8 Finding values Given a Probability In Section 5.2, we were given a normally distributed random variable x and we were asked to find a probability. In this section, we will be given a probability and we will be asked to find the value of the random variable x. 5.2 x z probability 5.3
9 Example: Finding a z-score Given an Area Find the z-score that corresponds to a cumulative area of z 0 z
10 Solution: Finding a z-score Given an Area Locate in the body of the Standard Normal Table. The z-score is The values at the beginning of the corresponding row and at the top of the column give the z-score.
11 Example: Finding a z-score Given an Area Find the z-score that has 10.75% of the distribution s area to its right = z z Because the area to the right is , the cumulative area is
12 Solution: Finding a z-score Given an Area Locate in the body of the Standard Normal Table. The z-score is The values at the beginning of the corresponding row and at the top of the column give the z-score.
13 Example: Finding a z-score Given an Area Find the z-score that has 96.16% of the distribution s area to its right. Find the z-score for which 95% of the distribution s area lies between z and z.
14 Review: Percentile P 83 : the 83 rd percentile, 83% of the data values below and 17% above P 21 : the 21 st percentile, 21% of the data values below and 79% above
15 Example: Finding a z-score Given a Percentile Find the z-score that corresponds to P z 0 The z-score that corresponds to P 5 is the same z- score that corresponds to an area of z The areas closest to 0.05 in the table are (z = 1.65) and (z = 1.64). Because 0.05 is halfway between the two areas in the table, use the z-score that is halfway between 1.64 and The z-score is
16 Example: Finding a z-score Given a Percentile P 50 P 90
17 Transforming a z-score to an x-score To transform a standard z-score to a data value x in a given population, use the formula z = x μ σ x = μ + zσ
18 Example: Finding an x-value A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to z-scores of 1.96, -0.44, and 0. z = 1.96: x = (2) = pounds z = -0.44: x = 9 + (-0.44)(2) = 8.12 pounds z = 0: x = 9 + 0(2) = 9 pounds Notice pounds is above the mean, 8.12 pounds is below the mean, and 9 pounds is equal to the mean.
19 Example: Finding a Specific Data Value Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score you can earn and still be eligible to be hired by the agency? An exam score in the top 10% is any score above the 90 th percentile. Find the z-score that corresponds to a cumulative area of 0.9.
20 Solution: Finding a Specific Data Value From the Standard Normal Table, the area closest to 0.9 is So the z-score that corresponds to an area of 0.9 is z = Using the equation x = μ + zσ x = (10) 62.8 The lowest score you can earn and still be eligible to be hired by the agency is about 63.
21 Example: Finding a Specific Data Value A researcher tests the braking distances of several cars. The braking distance from 60 miles per hour to a complete stop on dry pavement is measured in feet. The braking distances of a sample of cars are normally distributed, with a mean of 129 feet and a standard deviation of 5.18 feet. What is the longest braking distance one of these cars could have and still be in the bottom 1%?
22 Finding a Specific Data Value Using Graphing Calculator Inverse normal problem: 2 nd -> DISTR-> invnorm(area, μ, σ) If you use technology to find an x-value, it is not necessary first to find a z-score. For cumulative area of a specified z-score, μ = 0 and σ = 1. Practice with the previous example.
23 Example: Finding a Specific Data Value According to the United States Geological Survey, the mean magnitude of worldwide earthquakes in a recent year was about The magnitude of worldwide earthquakes can be approximated by a normal distribution. Assume the standard deviation is Between what two values does the middle 90% of the data lie?
24 Summary of Section 5.3 Calculate z from probability. Transform z to x.
25 Project 5 case study on P260 Each group has 4-5 people. Five groups in total. Work on all exercises in 30 minutes. One student will be assigned to present a solution of one question from each group. Each group submits one answer sheet.
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