Question: Find the percentage under the standard normal curve that is specified. 22) above \( z = -2.24 \)…

Find the percentage under the standard normal curve that is specified.

22. above \( z = -2.24 \)

23. below \( z = -2.24 \)

24. between \( z = 1.01 \) and \( z = 2.33 \)

25. between \( z = -02.17 \) and \( z = 0.4 \)

26. less than \( z = 1.98 \)

27. more than \( z = 2.46 \)

Solution

To solve these problems, we need to use the standard normal distribution (z-table). 22) Above \( z = -2.24 \) Find the probability that \( Z > -2.24 \). Since the standard normal distribution is symmetrical, \[ P(Z > -2.24) = 1 - P(Z < -2.24) \] Using the z-table, \( P(Z < -2.24) \approx 0.0125 \). \[ P(Z > -2.24) = 1 - 0.0125 = 0.9875 \] So the percentage is \( 98.75\% \). 23) Below \( z = -2.24 \) Find the probability that \( Z < -2.24 \). Using the z-table, \( P(Z < -2.24) \approx 0.0125 \). So the percentage is \( 1.25\% \). 24) Between \( z = 1.01 \) and \( z = 2.33 \) Find the probability that \( 1.01 < Z < 2.33 \). Using the z-table, \( P(Z < 1.01) \approx 0.8438 \) and \( P(Z < 2.33) \approx 0.9901 \). \[ P(1.01 < Z < 2.33) = P(Z < 2.33) - P(Z < 1.01) = 0.9901 - 0.8438 = 0.1463 \] So the percentage is \( 14.63\% \). 25) Between \( z = -2.17 \) and \( z = 0.4 \) Find the probability that \( -2.17 < Z < 0.4 \). Using the z-table, \( P(Z < -2.17) \approx 0.0150 \) and \( P(Z < 0.4) \approx 0.6554 \). \[ P(-2.17 < Z < 0.4) = P(Z < 0.4) - P(Z < -2.17) = 0.6554 - 0.0150 = 0.6404 \] So the percentage is \( 64.04\% \). 26) Less than \( z = 1.98 \) Find the probability that \( Z < 1.98 \). Using the z-table, \( P(Z < 1.98) \approx 0.9761 \). So the percentage is \( 97.61\% \). 27) More than \( z = 2.46 \) Find the probability that \( Z > 2.46 \). Since the standard normal distribution is symmetrical, \[ P(Z > 2.46) = 1 - P(Z < 2.46) \] Using the z-table, \( P(Z < 2.46) \approx 0.9931 \). \[ P(Z > 2.46) = 1 - 0.9931 = 0.0069 \] So the percentage is \( 0.69\% \).