Answer :
Final answer:
To find the probabilities in this normal distribution problem, we can use the Z-score formula. The Z-score formula is given by Z = (X - μ) / σ, where X is the value we are interested in, μ is the mean, and σ is the standard deviation. We can then use a Z-table or a calculator to find the probabilities and values associated with the Z-scores.
Explanation:
To find the probabilities in this normal distribution problem, we can use the Z-score formula. The Z-score formula is given by Z = (X - μ) / σ, where X is the value we are interested in, μ is the mean, and σ is the standard deviation.
a. To find the probability that X is less than 95, we need to calculate the Z-score for 95 using the formula: Z = (95 - 100) / 20 = -0.25. We can then use a Z-table or a calculator to find the probability associated with a Z-score of -0.25, which is approximately 0.401.
b. To find the probability that X is between 95 and 96.5, we need to calculate the Z-scores for the two values using the formula. The Z-score for 95 is -0.25 and the Z-score for 96.5 is (96.5 - 100) / 20 = -0.175. We can then subtract the probability associated with a Z-score of -0.175 from the probability associated with a Z-score of -0.25 to find the desired probability.
c. To find the probability that X is above 101.2, we need to calculate the Z-score for 101.2 using the formula: Z = (101.2 - 100) / 20 = 0.06. We can then use a Z-table or a calculator to find the probability associated with a Z-score of 0.06, which is approximately 0.523.
d. To find the value for X such that there is a 65% chance that X is above that value, we need to find the Z-score associated with a cumulative probability of 0.35 using a Z-table or calculator. Once we have the Z-score, we can use the formula Z = (X - μ) / σ to solve for X.
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