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XX is a normally distributed random variable with mean 4949 and standard deviation 2525.\newlineWhat is the probability that XX is between 2424 and 7474??\newlineUse the 0.68−0.95−0.9970.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.\newline____

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Q. XX is a normally distributed random variable with mean 4949 and standard deviation 2525.\newlineWhat is the probability that XX is between 2424 and 7474??\newlineUse the 0.68−0.95−0.9970.68-0.95-0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.\newline____
  1. Step 11: Calculate z-scores for X=24X = 24 and X=74X = 74: We have a normally distributed random variable XX with a mean (μ\mu) of 4949 and a standard deviation (σ\sigma) of 2525. We want to find the probability that XX is between 2424 and 7474. The X=74X = 7400-X=74X = 7411-X=74X = 7422 rule, also known as the empirical rule, states that for a normal distribution, approximately X=74X = 7433 of the data falls within one standard deviation of the mean, X=74X = 7444 within two standard deviations, and X=74X = 7455 within three standard deviations.
  2. Step 22: Interpret the z-scores: First, we calculate the z-scores for X=24X = 24 and X=74X = 74. The z-score is given by the formula z=X−μσz = \frac{X - \mu}{\sigma}. For X=24X = 24, the z-score is z=24−4925=−1z = \frac{24 - 49}{25} = -1. For X=74X = 74, the z-score is z=74−4925=1z = \frac{74 - 49}{25} = 1.
  3. Step 33: Apply the empirical rule: Since the z-scores for X=24X = 24 and X=74X = 74 are −1-1 and 11, respectively, this means that the values 2424 and 7474 are one standard deviation below and above the mean. According to the empirical rule, approximately 68%68\% of the data in a normal distribution falls within one standard deviation of the mean.
  4. Step 44: Determine the probability: Therefore, the probability that XX is between 2424 and 7474 is approximately 68%68\%, or 0.680.68 as a decimal.
  5. Step 55: Round the probability if necessary: We round the probability to the nearest thousandth if necessary. However, in this case, the probability is already at the desired precision, so no rounding is needed.

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