A normally distributed variable has a mean of 45. If 95% of the observations on the variable fall between 30 and 60, the standard deviation of the variable is approximately ________.
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A. B. C. D.Explanation
95% of the observations on a normally distributed variable lie within about two standard deviations of the mean. The standard deviation is then (45-30)/2 = 7.5.
To solve this problem, we need to use the concept of the empirical rule, also known as the 68-95-99.7 rule. This rule states that for a normally distributed variable, approximately:
In this case, we are given that 95% of the observations fall between 30 and 60. This means that the range of two standard deviations (2σ) covers 30 to 60, which implies that the mean plus two standard deviations (μ + 2σ) equals 60, and the mean minus two standard deviations (μ - 2σ) equals 30.
Let's solve for the standard deviation (σ):
μ + 2σ = 60 45 + 2σ = 60 2σ = 60 - 45 2σ = 15 σ = 15 / 2 σ ≈ 7.50
Therefore, the approximate standard deviation of the variable is 7.50. So, the correct answer is A. 7.50.