Final answer:
a) Probability x>75 is approximately 0.9938. b) Probability x<70 is approximately 0.0014. c) Probability x<80 or x>110 is approximately 0.1815. d) X values containing 80% of the values are between 87.2 and 112.8.
Step-by-step explanation:
a. To find the probability that x is greater than 75, we can use the standard normal distribution table or a calculator to find the z-score corresponding to 75 and then find the probability to the right of that value. The z-score for 75 is calculated as (75 - 100) / 10 = -2.5. Using the z-table or a calculator, we find that the probability to the right of -2.5 is approximately 0.9938.
b. To find the probability that x is less than 70, we can find the z-score for 70. The z-score for 70 is calculated as (70 - 100) / 10 = -3. Using the z-table or a calculator, we find that the probability to the left of -3 is approximately 0.0014.
c. To find the probability that x is less than 80 or greater than 110, we need to find the probability to the left of 80 and add it to the probability to the right of 110. The z-score for 80 is calculated as (80 - 100) / 10 = -2, and the z-score for 110 is calculated as (110 - 100) / 10 = 1. Using the z-table or a calculator, we find that the probability to the left of -2 is approximately 0.0228, and the probability to the right of 1 is approximately 0.1587. Adding these probabilities together, we get approximately 0.1815.
d. To find the X values that contain 80% of the values, we need to find the z-values that correspond to the percentile values of 10% and 90%. Using the z-table or a calculator, we find that the z-value for the 10th percentile is approximately -1.28, and the z-value for the 90th percentile is approximately 1.28. We can then use these z-values to find the X values by rearranging the z-score formula. The X value for the 10th percentile is calculated as -1.28 * 10 + 100 = 87.2, and the X value for the 90th percentile is calculated as 1.28 * 10 + 100 = 112.8. Therefore, 80% of the values are between 87.2 and 112.8.