X is a normally distributed random variable with mean 49 and standard deviation 25.What is the probability that X is between 24 and 74?Use the 0.68−0.95−0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.____
Q. X is a normally distributed random variable with mean 49 and standard deviation 25.What is the probability that X is between 24 and 74?Use the 0.68−0.95−0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.____
Step 1: Calculate z-scores for X=24 and X=74: We have a normally distributed random variable X with a mean (μ) of 49 and a standard deviation (σ) of 25. We want to find the probability that X is between 24 and 74. The X=740-X=741-X=742 rule, also known as the empirical rule, states that for a normal distribution, approximately X=743 of the data falls within one standard deviation of the mean, X=744 within two standard deviations, and X=745 within three standard deviations.
Step 2: Interpret the z-scores: First, we calculate the z-scores for X=24 and X=74. The z-score is given by the formula z=σX−μ. For X=24, the z-score is z=2524−49=−1. For X=74, the z-score is z=2574−49=1.
Step 3: Apply the empirical rule: Since the z-scores for X=24 and X=74 are −1 and 1, respectively, this means that the values 24 and 74 are one standard deviation below and above the mean. According to the empirical rule, approximately 68% of the data in a normal distribution falls within one standard deviation of the mean.
Step 4: Determine the probability: Therefore, the probability that X is between 24 and 74 is approximately 68%, or 0.68 as a decimal.
Step 5: 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.
More problems from Find probabilities using the normal distribution I