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XX is a normally distributed random variable with mean 6565 and standard deviation 88.\newlineWhat is the probability that XX is between 6262 and 6868??\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____

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Q. XX is a normally distributed random variable with mean 6565 and standard deviation 88.\newlineWhat is the probability that XX is between 6262 and 6868??\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____
  1. Convert X values to Z-scores: To find the probability that X is between 6262 and 6868, 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=(Xμ)σZ = \frac{(X - \mu)}{\sigma}, where XX is the value, μ\mu is the mean, and σ\sigma is the standard deviation.
  2. Calculate Z-score for lower bound: Let's calculate the Z-score for the lower bound, which is X1=62X_1 = 62.\newlineUsing the formula Z=XμσZ = \frac{X - \mu}{\sigma}, we get Z1=62658=38=0.375Z_1 = \frac{62 - 65}{8} = \frac{-3}{8} = -0.375.
  3. Calculate Z-score for upper bound: Now, let's calculate the Z-score for the upper bound, which is X2=68X_2 = 68.\newlineUsing the formula Z=XμσZ = \frac{X - \mu}{\sigma}, we get Z2=68658=38=0.375Z_2 = \frac{68 - 65}{8} = \frac{3}{8} = 0.375.
  4. Find probability for Z1Z_1 and Z2Z_2: Next, we need to find the probability that ZZ is less than Z1Z_1 and Z2Z_2. We can use the standard normal distribution table or a calculator with normal distribution functions to find these probabilities. The probability for Z1=0.375Z_1 = -0.375 is approximately P(Z < -0.375) = 0.3531, and the probability for Z2=0.375Z_2 = 0.375 is approximately P(Z < 0.375) = 0.6463.
  5. Calculate probability of X between 6262 and 6868: To find the probability that X is between 6262 and 6868, we need to subtract the probability of Z1Z_1 from the probability of Z2Z_2. This gives us P(62 < X < 68) = P(Z_2) - P(Z_1) = 0.6463 - 0.3531 = 0.2932.

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