Final answer:
To find the values of xo for different probabilities, we can use the standard normal distribution table or a calculator to calculate the cumulative probability function. For example, to find xo such that P(x < xo) = 0.8413, we use the z-score formula: xo = mu + (sigma * z-score). The answers for each part are: a. xo = 53.9835, b. xo = 56.88, c. xo = 44.065, d. xo = 51.5798, e. xo = 45.154, and f. xo = 56.978.
Step-by-step explanation:
a. P(x < xo) = 0.8413. To find xo, we need to find the z-score corresponding to this probability. Using a standard normal distribution table, we find that the z-score for a cumulative probability of 0.8413 is approximately 0.9945. We can then use the z-score formula to find xo: xo = mu + (sigma * z-score) = 50 + (3 * 0.9945) = 53.9835.
b. P(x > xo) = 0.025. To find xo, we need to find the z-score corresponding to this probability. Using a standard normal distribution table, we find that the z-score for a cumulative probability of 0.975 is approximately 1.96. We can then use the z-score formula to find xo: xo = mu + (sigma * z-score) = 50 + (3 * 1.96) = 56.88.
c. P(x > xo) = 0.95. To find xo, we need to find the z-score corresponding to this probability. Using a standard normal distribution table, we find that the z-score for a cumulative probability of 0.05 is approximately -1.645. We can then use the z-score formula to find xo: xo = mu + (sigma * z-score) = 50 + (3 * -1.645) = 44.065.
d. P(41 < x < xo) = 0.8630. We can subtract the cumulative probability of x < 41 from the cumulative probability of x < xo to get the probability of 41 < x < xo. Using a standard normal distribution table, we find that the cumulative probability of x < 41 is approximately 0.1611. We can then calculate the probability of 41 < x < xo: 0.8630 - 0.1611 = 0.7019. To find xo, we need to find the z-score corresponding to this probability. Using a standard normal distribution table, we find that the z-score for a cumulative probability of 0.7019 is approximately 0.5266. We can then use the z-score formula to find xo: xo = mu + (sigma * z-score) = 50 + (3 * 0.5266) = 51.5798.
e. 10% of the values of x are less than xo. We can find the z-score corresponding to a cumulative probability of 0.10, which is approximately -1.282. We can then use the z-score formula to find xo: xo = mu + (sigma * z-score) = 50 + (3 * -1.282) = 45.154.
f. 1% of the values of x are greater than xo. We can find the z-score corresponding to a cumulative probability of 0.99, which is approximately 2.326. We can then use the z-score formula to find xo: xo = mu + (sigma * z-score) = 50 + (3 * 2.326) = 56.978.