4. Suppose the random variable X𝑋 is normally distributed with μ=50𝜇=50 and σ2=16𝜎2=16 . What is P(49≤X≤55)P(49≤𝑋≤55) ?Multiple choice 2 Question 3 0.3756 0.4931 0.6734 0.2098
Question
4. Suppose the random variable X𝑋 is normally distributed with μ=50𝜇=50 and σ2=16𝜎2=16 . What is P(49≤X≤55)P(49≤𝑋≤55) ?Multiple choice 2 Question 3 0.3756 0.4931 0.6734 0.2098
Solution
To solve this problem, we need to convert the raw scores to z-scores and then look up the probabilities in the z-table (standard normal distribution table).
Step 1: Convert the raw scores to z-scores. The formula for converting a raw score x to a z-score is z = (x - μ) / σ.
For x = 49, z = (49 - 50) / sqrt(16) = -0.25 For x = 55, z = (55 - 50) / sqrt(16) = 1.25
Step 2: Look up the probabilities in the z-table.
The probability for z = -0.25 is 0.4013 (this is the probability to the left of the z-score, i.e., P(X < 49)). The probability for z = 1.25 is 0.8944 (this is the probability to the left of the z-score, i.e., P(X < 55)).
Step 3: Subtract the probabilities to get P(49 ≤ X ≤ 55).
P(49 ≤ X ≤ 55) = P(X < 55) - P(X < 49) = 0.8944 - 0.4013 = 0.4931
So, the answer is 0.4931.
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