A bell-shaped, symmetrical frequency distribution 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:
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A. B. C. D.B
95% of the observations in a bell-shaped, symmetrical frequency distribution lie within approximately 2 standard deviations of the mean. The standard deviation is then (45-30)/2 = 7.50.
To find the standard deviation of the variable in a bell-shaped, symmetrical frequency distribution, we can use the empirical rule, also known as the 68-95-99.7 rule. This rule states that in a normal distribution:
In this case, we are given that 95% of the observations fall between 30 and 60. Since 95% falls within two standard deviations, we can deduce that the distance between the mean and either boundary is equal to two standard deviations.
Let's calculate the standard deviation step by step:
Determine the range: The range is the difference between the upper and lower boundaries, which in this case is 60 - 30 = 30.
Divide the range by 4: To find the standard deviation, we divide the range by 4, because two standard deviations cover 95% of the observations. Therefore, 30 / 4 = 7.5.
Hence, the standard deviation of the variable is 7.5.
Therefore, the correct answer is B. 7.50.