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Answer:
a. To find P(X ≤ 100), we need to calculate the cumulative distribution function (CDF) of the normal distribution with mean μ = 102 and standard deviation σ = 34 at the value x = 100. Using the standard normal distribution table or a calculator, we can find the z-score corresponding to x = 100: z = (x - μ) / σ = (100 - 102) / 34 = -2 / 34 = -0.0588 (rounded to 4 decimal places) Next, we find the area to the left of this z-score on the standard normal distribution table, which represents the probability: P(X ≤ 100) = P(Z ≤ -0.0588) Using the standard normal distribution table or a calculator, we find that the probability is approximately 0.4783 (rounded to 4 decimal places). Therefore, P(X ≤ 100) ≈ 0.4783. b. To find P(95 ≤ X ≤ 110), we need to calculate the cumulative distribution function (CDF) of the normal distribution with mean μ = 102 and standard deviation σ = 34 at the values x = 95 and x = 110. First, we calculate the z-scores for x = 95 and x = 110: z1 = (95 - 102) / 34 = -0.2059 (rounded to 4 decimal places) z2 = (110 - 102) / 34 = 0.2353 (rounded to 4 decimal places) Next, we find the area between these two z-scores on the standard normal distribution table, which represents the probability: P(95 ≤ X ≤ 110) = P(-0.2059 ≤ Z ≤ 0.2353) Using the standard normal distribution table or a calculator, we find that the probability is approximately 0.3982 (rounded to 4 decimal places). Therefore, P(95 ≤ X ≤ 110) ≈ 0.3982. c. To find x such that P(X ≤ x) = 0.360, we need to find the z-score that corresponds to this probability. Using the standard normal distribution table or a calculator, we find the z-score that corresponds to a probability of 0.360: z = -0.3365 (rounded to 4 decimal places) Next, we use the z-score formula to find x: z = (x - μ) / σ Solving for x, we have: x = z * σ + μ = -0.3365 * 34 + 102 = 90.563 (rounded to 3 decimal places) Therefore, x ≈ 90.563.