Explanation Approach: The bell-shaped normal curve is symmetric with respect to its mean (denoted as μ ). Thus, 50% of the data values lies to the left of mean and remaining 50% of the data lies to the right of the mean. Figure 1 illustrates 68-95-99.7 rule for a normal distribution . Figure 1 Approximately 68% of the data lies between μ − σ and μ + σ , 95% of the data lies between μ − 2 σ and μ + 2 σ , 99.7% of the data lies between μ − 3 σ and μ + 3 σ . Assume a normal distribution has a mean of μ and a standard deviation of σ . We use the equation z = x − μ σ to convert a value x in the nonstandard distribution to a z -score. Given Information: The mean of the distribution of lengths of human pregnancies is 268 days. Also, 95% of human pregnancies last between 250 days and 286 days. Calculation: In normal distribution, 95% of the data lies between μ − 2 σ and μ + 2 σ . Given that 95% of all human pregnancies last between 250 days and 286 days. Therefore, μ − 2 σ = 250 and μ + 2 σ = 286. Substitute μ = 268 in any of the above two equations and simplify. 268 − 2 σ = 250 2 σ = 268 − 250 2 σ = 18 σ = 18 2 = 9 Now, substitute μ = 268 , σ = 9 and x = 275 in z = x − μ σ , z = 275 − 268 9 = 7 9 ≈ 0.78 Therefore, the corresponding z -score is approximately 0.78. That is, P ( x ≥ 275 ) = P ( z ≥ 0.78 ) . Figure 1 To find the area between z = 0 and to a z -score, use the data in the standard normal distribution table. Use the row for z = 0

6th Edition

ISBN: 9780134434681

Author: Tom Pirnot

Publisher: PEARSON

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 14.4, Problem 57E

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The standard deviation of the distribution and the percent of human pregnancies expected to last at least 275 days.

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Chapter 14.4, Problem 56E

Chapter 14.4, Problem 58E

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Chapter 14 Solutions

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The standard deviation of the distribution and the percent of human pregnancies expected to last at least 275 days.

Solution Summary: The author illustrates the 68-95-99.7 rule for a normal distribution. The bell-shaped normal curve is symmetric with respect to its mean.

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To find: The standard deviation of the distribution and the percent of human pregnancies expected to last at least 275 days.

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Chapter 14 Solutions

To find:

The standard deviation of the distribution and the percent of human pregnancies expected to last at least 275 days.