X is a normally distributed random variable with mean 65 and standard deviation 8.What is the probability that X is between 62 and 68?Write your answer as a decimal rounded to the nearest thousandth.____
Q. X is a normally distributed random variable with mean 65 and standard deviation 8.What is the probability that X is between 62 and 68?Write your answer as a decimal rounded to the nearest thousandth.____
Convert X values to Z-scores: To find the probability that X is between 62 and 68, we first need to convert the X values to Z-scores, which are standardized scores that tell us how many standard deviations away from the mean our values are. The formula for a Z-score is Z=σ(X−μ), where X is the value, μ is the mean, and σ is the standard deviation.
Calculate Z-score for lower bound: Let's calculate the Z-score for the lower bound, which is X1=62.Using the formula Z=σX−μ, we get Z1=862−65=8−3=−0.375.
Calculate Z-score for upper bound: Now, let's calculate the Z-score for the upper bound, which is X2=68.Using the formula Z=σX−μ, we get Z2=868−65=83=0.375.
Find probability for Z1 and Z2: Next, we need to find the probability that Z is less than Z1 and Z2. We can use the standard normal distribution table or a calculator with normal distribution functions to find these probabilities. The probability for Z1=−0.375 is approximately P(Z < -0.375) = 0.3531, and the probability for Z2=0.375 is approximately P(Z < 0.375) = 0.6463.
Calculate probability of X between 62 and 68: To find the probability that X is between 62 and 68, we need to subtract the probability of Z1 from the probability of Z2. This gives us P(62 < X < 68) = P(Z_2) - P(Z_1) = 0.6463 - 0.3531 = 0.2932.
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