One way to measure the width of a piece of paper is with a ruler. The distribution of measurement error (measured width - actual width) using this method is approximately normal with mean 0 millimeters and standard deviation 1.2 millimeters. About what proportion of all measurements of this piece of paper are within 0.3 millimeters of the actual width? (A) 0.0987 (B) 0.1974 (C) 0.2500 (D) 0.7500 (E) 0.8026
Question
One way to measure the width of a piece of paper is with a ruler. The distribution of measurement error (measured width - actual width) using this method is approximately normal with mean 0 millimeters and standard deviation 1.2 millimeters. About what proportion of all measurements of this piece of paper are within 0.3 millimeters of the actual width? (A) 0.0987 (B) 0.1974 (C) 0.2500 (D) 0.7500 (E) 0.8026
Solution
To answer this question, we need to use the properties of the standard normal distribution.
First, we need to standardize the value of 0.3 millimeters using the given standard deviation of 1.2 millimeters. This is done by dividing 0.3 by 1.2, which gives us 0.25.
Next, we look up this value in the standard normal distribution table or use a calculator with a normal distribution function. The value we get is the proportion of measurements that are within 0.25 standard deviations of the mean, which is 0.5987.
However, this value includes both measurements that are 0.25 standard deviations above the mean and 0.25 standard deviations below the mean. Since we are only interested in measurements that are within 0.3 millimeters of the actual width, we need to consider only half of this proportion.
So, we divide 0.5987 by 2, which gives us 0.29935.
Finally, since the distribution is symmetric around the mean, the proportion of measurements that are within 0.3 millimeters of the actual width is twice this value, or 0.5987.
So, the answer is (E) 0.8026.
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