Chapter 11 Normal Distributions
Back to Lesson 11-2
Chapter 11 Normal Distributions
6. Using a statistics utility we find the probability to be about 0.6450.
Lesson 11-2(pp. 692-697) 1. The mean of the standard normal distribution is 0 because it has y-axis symmetry, and the standard deviation is 1, by definition. 2. a. Using the standard normal table, we find the area below 1.5 to be about 0.9332. b. The total area under the curve is 1 so the area to the right of the shaded area under the curve is given by 1 – 0.9332 = 0.0668. 3. a. The area under the curve below 1.5 is 0.9332, and the area under the curve below 0 is 0.5, so the area between 0 and 1.5 is 0.9332 – 0.5 = 0.4332. b. This is the probability that z is between -1.5 and 1.5. This is twice the area between 0 and 1.5 under the curve. This is 2(0.4332) = 0.8664. c. The probability that z is greater than -1.5 is the same (by symmetry) as the probability that z is less than 1.5, which is 0.9332. 4. Using a statistics utility we find that the probability is about 0.9265.
5. Using a statistics utility we find the probability to be about 0.4761.
315 Functions, Statistics, and Trigonometry Solution Manual
7. Using a statistics utility we find the probability to be about 0.9987.
8. Using a statistics utility we find the probability to be about 0.4515. 9. Using the standard normal table we find the probability that z is less than 2 to be 0.9772. Because of this, we know the probability that z is greater than 2 is 0.0228. By symmetry, we know the probability that z is less than -2 is also 0.0228, so the probability that z is between -2 and 2 is 0.9772 – 0.0228 = 0.9545, or about 95.45%. 10. Using the standard normal table we find the probability that z is less than 3 to be 0.9987. Because of this, we know the probability that z is greater than 3 is 0.0013. By symmetry, we know the probability that z is less than -3 is also 0.0013, so the probability that z is between -3 and 3 is 0.9987 – 0.0013 = 0.9974, or about 99.7%. 11. The probability that z is less than 1 is 0.8413. The probability that z is less than -1 is the same as the probability that z is greater than 1, which is 1 – 0.8413. This makes the probability that z is between -1 and 1 equal to 0.8413 – (1 – 0.8413) = 0.6826. Chapter 11, Lesson 11-2
Back to Lesson 11-2
12. The values in the table are all above 0.5, so we will not find 0.1736 on the standard normal table. By symmetry, 1 – 0.1736 = 0.8264 is on the table, which will have the opposite sign of the z-score in question. This corresponds to 0.94. Thus the value we are looking for is -0.94. 13. The standard normal table shows the probability that z is less than some value c. The probability that z is greater than c is 1 – f(z). Thus, 0.2514 = 1 – f(z), so f(z) = 0.7486. We can find this value on the standard normal table; it corresponds to z = 0.67. 14. The value that is exceeded by 25% so about 75% will be below the value. This means we are looking for the value of z such that f(z) = 0.75. We find that this corresponds to about 0.67. 15. We know that if 90% of observations fall within a certain distance of the mean, that 5% of values must fall above that distance above the mean. This means we are looking for the z whose f(z) is about 0.95. This corresponds to about 1.645.
20. To find the z-score, subtract the mean and divide by the standard deviation. = -1.4 . This gives 6875 5 21. a. To find the mean, add the values and divide by the number of values. This gives x = 122. To find the standard deviation, subtract the mean from each value, square each deviation, add them together, and take the square root. This gives s = 24.6475. b. Apply the transformation to each data point and follow the process from part a. This gives x = 0 and s = 1. This makes sense because the mean of the standard normal curve is 0 and the standard deviation is 1. 22. a. i. 0.5140 ii. 0.5271 b. The graph of the standard normal curve is nearly a straight line (linear) on the interval 0 < z < 0.09. c. The graph of the standard normal curve is not close to linear in other areas.
16. We are looking for the percentage of people who get over 700, which is the same as the area under the normal curve above z = 2. This corresponds to 1 – f(2) = 1 – 0.9772 = 0.0228, or about 2.28%. 17. The area under the parent normal curve is , while the area under the standard normal curve is 1. 18. False; we can see by looking at the parent normal curve that the area under the parent normal curve is not equal to 1. 19. We know that μ = 40 = np , and
= 4.6 = np(1 p) . Solving these equations gives n = 85 and p 0.471 . 316 Functions, Statistics, and Trigonometry Solution Manual
Chapter 11, Lesson 11-2
Chapter 11 Normal Distributions
6. Using a statistics utility we find the probability to be about 0.6450.
Lesson 11-2(pp. 692-697) 1. The mean of the standard normal distribution is 0 because it has y-axis symmetry, and the standard deviation is 1, by definition. 2. a. Using the standard normal table, we find the area below 1.5 to be about 0.9332. b. The total area under the curve is 1 so the area to the right of the shaded area under the curve is given by 1 – 0.9332 = 0.0668. 3. a. The area under the curve below 1.5 is 0.9332, and the area under the curve below 0 is 0.5, so the area between 0 and 1.5 is 0.9332 – 0.5 = 0.4332. b. This is the probability that z is between -1.5 and 1.5. This is twice the area between 0 and 1.5 under the curve. This is 2(0.4332) = 0.8664. c. The probability that z is greater than -1.5 is the same (by symmetry) as the probability that z is less than 1.5, which is 0.9332. 4. Using a statistics utility we find that the probability is about 0.9265.
5. Using a statistics utility we find the probability to be about 0.4761.
315 Functions, Statistics, and Trigonometry Solution Manual
7. Using a statistics utility we find the probability to be about 0.9987.
8. Using a statistics utility we find the probability to be about 0.4515. 9. Using the standard normal table we find the probability that z is less than 2 to be 0.9772. Because of this, we know the probability that z is greater than 2 is 0.0228. By symmetry, we know the probability that z is less than -2 is also 0.0228, so the probability that z is between -2 and 2 is 0.9772 – 0.0228 = 0.9545, or about 95.45%. 10. Using the standard normal table we find the probability that z is less than 3 to be 0.9987. Because of this, we know the probability that z is greater than 3 is 0.0013. By symmetry, we know the probability that z is less than -3 is also 0.0013, so the probability that z is between -3 and 3 is 0.9987 – 0.0013 = 0.9974, or about 99.7%. 11. The probability that z is less than 1 is 0.8413. The probability that z is less than -1 is the same as the probability that z is greater than 1, which is 1 – 0.8413. This makes the probability that z is between -1 and 1 equal to 0.8413 – (1 – 0.8413) = 0.6826. Chapter 11, Lesson 11-2
Back to Lesson 11-2
12. The values in the table are all above 0.5, so we will not find 0.1736 on the standard normal table. By symmetry, 1 – 0.1736 = 0.8264 is on the table, which will have the opposite sign of the z-score in question. This corresponds to 0.94. Thus the value we are looking for is -0.94. 13. The standard normal table shows the probability that z is less than some value c. The probability that z is greater than c is 1 – f(z). Thus, 0.2514 = 1 – f(z), so f(z) = 0.7486. We can find this value on the standard normal table; it corresponds to z = 0.67. 14. The value that is exceeded by 25% so about 75% will be below the value. This means we are looking for the value of z such that f(z) = 0.75. We find that this corresponds to about 0.67. 15. We know that if 90% of observations fall within a certain distance of the mean, that 5% of values must fall above that distance above the mean. This means we are looking for the z whose f(z) is about 0.95. This corresponds to about 1.645.
20. To find the z-score, subtract the mean and divide by the standard deviation. = -1.4 . This gives 6875 5 21. a. To find the mean, add the values and divide by the number of values. This gives x = 122. To find the standard deviation, subtract the mean from each value, square each deviation, add them together, and take the square root. This gives s = 24.6475. b. Apply the transformation to each data point and follow the process from part a. This gives x = 0 and s = 1. This makes sense because the mean of the standard normal curve is 0 and the standard deviation is 1. 22. a. i. 0.5140 ii. 0.5271 b. The graph of the standard normal curve is nearly a straight line (linear) on the interval 0 < z < 0.09. c. The graph of the standard normal curve is not close to linear in other areas.
16. We are looking for the percentage of people who get over 700, which is the same as the area under the normal curve above z = 2. This corresponds to 1 – f(2) = 1 – 0.9772 = 0.0228, or about 2.28%. 17. The area under the parent normal curve is , while the area under the standard normal curve is 1. 18. False; we can see by looking at the parent normal curve that the area under the parent normal curve is not equal to 1. 19. We know that μ = 40 = np , and
= 4.6 = np(1 p) . Solving these equations gives n = 85 and p 0.471 . 316 Functions, Statistics, and Trigonometry Solution Manual
Chapter 11, Lesson 11-2
