76 Chapter 6 Continuous Random Variables and the Normal Distribution

Continuous Random Variables and the Normal Distribution____________

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Section 6.5 Example 6-12, pg. 271 Finding Area under a Normal Curve The assembly time for a toy racing car follows a normal distribution with a mean of 55 minutes and a standard deviation of 4 minutes. What is the probability that an assembly worker will finish assembling a racing car in 60 minutes? For this example, you will use MINITAB to calculate P( X ≤ 60) for the normal random variable, X, with a mean of 55 minutes and a standard deviation of 4 minutes. To do this in MINITAB, click on Calc → Probability Distributions → Normal. On the input screen, select Cumulative probability. (Cumulative probability 'accumulates' all probability to the left of a specific value, which in this case is 60.) Enter 55 for the Mean and 4 for the Standard deviation. Next select Input Constant and enter the value 60.

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Click on OK and the probability should be displayed in the Session Window.

The probability that the worker will complete the toy in 60 minutes is 0.8944.

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78 Chapter 6 Continuous Random Variables and the Normal Distribution

Example 6-13, pg. 272 Finding Area under a Normal Curve The amount of soda a filling machine pours into 12 ounce cans is normally distributed with µ=12 ounces and σ=.015 ounces. What is the probability that a randomly selected can contains between 11.97 and 11.99 ounces of soda? To find P(11.97 ≤ X ≤ 11.99) using MINITAB, you will need to calculate two probabilities: P(X≤11.97) and P(X≤11.99). These two calculations are required because MINITAB calculates the accumulated probability to the left of a given value. The first calculation, P(X≤11.97) calculates the probability to the left of 11.97 and the second calculation calculates the probability to the left of 11.99. Click on Calc → Probability Distributions → Normal. On the input screen, select Cumulative probability. Enter 12 for the Mean and .015 for the Standard deviation. Next select Input Constant and enter the value 11.97. Click on OK. Repeat the above steps using an Input constant of 11.99. The Session Window should have both results: P(X ≤ 11.97) and P(X ≤ 11.99).

So, to find the probability between 11.97 and 11.99, that is, P(11.97 ≤ X ≤ 11.99), you must subtract the two probabilities. Thus, the P(11.97 ≤ X ≤ 11.99) = .2525 - .0228 = .2297. (Note: Minitab uses more decimal places than the textbook does so answers will vary slightly.)

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Section 6.6 Example 6-18, pg. 277 Finding a specific data value The life of a calculator has a normal distribution with a mean of 54 months and a standard deviation of 8 months. What should the warranty period be to replace a malfunctioning calculator if the company does not want to replace more than 1% of all the calculators sold? In this example, you are asked to find the specific X-value that marks the bottom 1% of the distribution. To do this in MINITAB, click on Calc → Probability Distributions → Normal. On the input screen, select Inverse Cumulative probability. Enter 54 months for the Mean and 8 months for the Standard deviation. For this type of problem, the Input constant will be the area to the left of the X-value we are looking for. This input constant will be a decimal number between 0 and 1. Select Input Constant and enter the value .01 which represents the bottom 1% of the distribution. Click on OK and the X-value should be in the Session Window.

The x-value that represents the cut-off point for the bottom 1% of the distribution is 35.39. Notice that this value differs slightly from the textbook answer. The difference is due to rounding.

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Example 6-19, pg. 278 Finding a specific data value SAT scores have a normal distribution with a mean of 1020 and a standard deviation of 153. What score would a student need to attain so that only 10% of all examinees score higher than he/she does? In this example, you are asked to find the specific X-value that marks the top 10% of the distribution. To do this in MINITAB, click on Calc → Probability Distributions → Normal. On the input screen, select Inverse Cumulative probability. Enter 1020 for the Mean and 153 for the Standard deviation. The Input constant will be the area to the left of the X-value we are looking for. In this example, the Input constant is .90, since you are looking for the X-value that separates the top 10% from the bottom 90%. Select Input Constant and enter the value .90 which represents the bottom 90% of the distribution. Click on OK and the X-value should be in the Session Window.

The x-value that represents the cut-off point for the top 10% of the distribution is 1216.08.

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82 Chapter 6 Continuous Random Variables and the Normal Distribution

Suggested Exercises Section 6.5 pp. 274 – 275: 6.41, 6.43, 6.51 (Note: Minitab instructions for italicized exercises appear below.) Exercise 6.41(a), pg.274 For this example, you will use MINITAB to calculate P( X > 40) for the normal random variable, X, with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. To do this in MINITAB, click on Calc → Probability Distributions → Normal. On the input screen, select Cumulative probability. (Cumulative probability 'accumulates' all probability to the left of a specific value, which in this case is 40.) Enter 46 for the Mean and 4 for the Standard deviation. Next select Input Constant and enter the value 40.

What you see in the Session Window is P( X ≤ 40) . To calculate P( X > 40) , simply subtract the displayed value from 1 since P( X > 40) = 1- P( X ≤ 40) . The correct answer is 1-.0668, or .9332. Æ

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Section 6.6 pp. 279 – 280: 6.59, 6.61 Mini-Projects p. 292: 6-1, 6-2 (Note: Minitab instructions for italicized exercises appear below.) Mini-Project 6-1: Open the NBA dataset. a. To draw a histogram of the heights, click on Stat → Basic Statistics → Display Descriptive Statistics. Select C2 for the Variable and click on Graphs. Select Histogram of Data and click on OK twice to view the histogram.

b. The descriptive statistics for the height data are in the Session Window and can be seen after the Graph Window is closed. (Note: The standard deviation that Minitab calculates is the sample standard deviation, s, not the population standard deviation, σ. For a large dataset, such as this one, there is very little difference between s and σ so you can use s as an approximation of σ. c. Using a calculator, calculate the intervals: µ±σ, µ±2σ and µ±3σ. To find the percentage of the data that falls within each of these intervals, you must sort the data. To do this, click on Data → Sort. On the input screen that appears, Sort columns: C1-C3, sort By column: C2, and for Store sorted data in:, select Columns of current worksheet and enter C4- C6. The original data is now sorted by height and is stored in C4 – C6.

84 Chapter 6 Continuous Random Variables and the Normal Distribution

Scroll through the height data in C5 and count the number of data points that fall in the intervals: µ±σ, µ±2σ and µ±3σ. Æ

Technology Assignments p. 295: TA6.3, TA6.4, TA6.5