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Section 7.1 Graphs of Normal Probability Distributions

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Properties of The Normal Distribution

The curve is bell-shaped with the highest point over the mean, μ. 2

Properties of The Normal Distribution

The curve is symmetrical about a vertical line through μ. 7.1 / 3

Properties of The Normal Distribution

The curve approaches the horizontal axis but never touches or crosses it. 7.1 / 4

Properties of The Normal Distribution

The transition points between cupping upward and downward occur above μ + σ and μ – σ . 7.1 / 5

Properties of The Normal Distribution • The curve is bell-shaped with the highest point over the mean, μ. • The curve is symmetrical about a vertical line through μ. • The curve approaches the horizontal axis but never touches or crosses it. The transition points between cupping upward and downward occur above μ + σ and μ – σ . 7.1 / 6

The Empirical Rule Applies to any symmetrical and bell-shaped distribution

Approximately 68% of the data values lie within one standard deviation of the mean.

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The Empirical Rule Approximately 95% of the data values lie within two standard deviations of the mean.

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The Empirical Rule Almost all (approximately 99.7%) of the data values will be within three standard deviations of the mean.

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Application of the Empirical Rule The life of a particular type of light bulb is normally distributed with a mean of 1100 hours and a standard deviation of 100 hours. What is the probability that a light bulb of this type will last between 1000 and 1200 hours? Approximately 68%

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Example Male adult heights in North America are approximately normally distributed with a mean of 70 inches and standard deviation of 4 inches. Find the height that is a. 3 standard deviations above the mean b. 2 standard deviations below the mean c. The percentage of men with heights Between 66 inches and 74 inches d. The percentage of men with heights Between 70 inches and 74 inches e. The percentage of men with heights above 78 inches. Solution a. 82 inches b. 62 inches c. 68% d. 34% e. 2.5% Assignment 13

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Normal Probability Distribution (calc) • The Normal Probability Distribution menu for the TI-83+/84+ is found under DISTR (2nd VARS).

• NOTE: A mean of zero and a standard deviation of one are considered to be the default values for a normal distribution on the calculator, if you choose not to set these values. • • The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve. Using this function returns the y-coordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation)

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Normal Probability Distribution (calc) #2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) #3: invNorm( inv = Inverse Normal Probability Distribution Function This function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invNorm (probability, mean, standard deviation) 7.1 / 13

Example • Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90th percentile for this distribution. (answers will be rounded to the nearest thousandth) 1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.) Syntax: normalcdf(lower bound, upper bound, mean, standard deviation) • Answer: The probability is 85.361%. 7.1 / 14

Example cont. 1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest value the calculator can handle is 1 x 1099. Type 1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) Answer: The probability is 13.326%. 1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest value the calculator can handle is -1 x 1099. Type -1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) • Answer: The probability is 3.772%.

1d: Find the 90th percentile for this distribution. (Given a probability region to the left of a value (i.e., a percentile), determine the value using invNorm.) • Answer: The x-value is 75.767.

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7.1 / 1

Properties of The Normal Distribution

The curve is bell-shaped with the highest point over the mean, μ. 2

Properties of The Normal Distribution

The curve is symmetrical about a vertical line through μ. 7.1 / 3

Properties of The Normal Distribution

The curve approaches the horizontal axis but never touches or crosses it. 7.1 / 4

Properties of The Normal Distribution

The transition points between cupping upward and downward occur above μ + σ and μ – σ . 7.1 / 5

Properties of The Normal Distribution • The curve is bell-shaped with the highest point over the mean, μ. • The curve is symmetrical about a vertical line through μ. • The curve approaches the horizontal axis but never touches or crosses it. The transition points between cupping upward and downward occur above μ + σ and μ – σ . 7.1 / 6

The Empirical Rule Applies to any symmetrical and bell-shaped distribution

Approximately 68% of the data values lie within one standard deviation of the mean.

7

The Empirical Rule Approximately 95% of the data values lie within two standard deviations of the mean.

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The Empirical Rule Almost all (approximately 99.7%) of the data values will be within three standard deviations of the mean.

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Application of the Empirical Rule The life of a particular type of light bulb is normally distributed with a mean of 1100 hours and a standard deviation of 100 hours. What is the probability that a light bulb of this type will last between 1000 and 1200 hours? Approximately 68%

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Example Male adult heights in North America are approximately normally distributed with a mean of 70 inches and standard deviation of 4 inches. Find the height that is a. 3 standard deviations above the mean b. 2 standard deviations below the mean c. The percentage of men with heights Between 66 inches and 74 inches d. The percentage of men with heights Between 70 inches and 74 inches e. The percentage of men with heights above 78 inches. Solution a. 82 inches b. 62 inches c. 68% d. 34% e. 2.5% Assignment 13

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Normal Probability Distribution (calc) • The Normal Probability Distribution menu for the TI-83+/84+ is found under DISTR (2nd VARS).

• NOTE: A mean of zero and a standard deviation of one are considered to be the default values for a normal distribution on the calculator, if you choose not to set these values. • • The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve. Using this function returns the y-coordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation)

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Normal Probability Distribution (calc) #2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) #3: invNorm( inv = Inverse Normal Probability Distribution Function This function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invNorm (probability, mean, standard deviation) 7.1 / 13

Example • Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90th percentile for this distribution. (answers will be rounded to the nearest thousandth) 1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.) Syntax: normalcdf(lower bound, upper bound, mean, standard deviation) • Answer: The probability is 85.361%. 7.1 / 14

Example cont. 1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest value the calculator can handle is 1 x 1099. Type 1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) Answer: The probability is 13.326%. 1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest value the calculator can handle is -1 x 1099. Type -1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.) • Answer: The probability is 3.772%.

1d: Find the 90th percentile for this distribution. (Given a probability region to the left of a value (i.e., a percentile), determine the value using invNorm.) • Answer: The x-value is 75.767.

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