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Re: The length of bolts made in factory Z is normally distribute [#permalink]
sandy wrote:
The length of bolts made in factory Z is normally distributed, with a mean length of 0.1630 meters and a standard deviation of 0.0084 meters. The probability that a randomly selected bolt is between 0.1546 meters and 0.1756 meters long is between

(A) 54% and 61%.
(B) 61% and 68%.
(C) 68% and 75%.
(D) 75% and 82%.
(E) 82% and 89%.



Better if it is explained by the picture to picture.
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Re: The length of bolts made in factory Z is normally distribute [#permalink]
1
sandy wrote:
The length of bolts made in factory Z is normally distributed, with a mean length of 0.1630 meters and a standard deviation of 0.0084 meters. The probability that a randomly selected bolt is between 0.1546 meters and 0.1756 meters long is between

(A) 54% and 61%.
(B) 61% and 68%.
(C) 68% and 75%.
(D) 75% and 82%.
(E) 82% and 89%.



Official Explanation:

First, make the numbers easier to use. Either multiply every number by the same constant or
move the decimal the same number of places for each number. In the case of moving the decimal four
places, the mean becomes 1,630, the standard deviation becomes 84, and the two other numbers
become 1,546 and 1,756.

Next, “normalize” the boundaries. That is, take 1,546 meters (the lower boundary) and 1,756 meters
(the upper boundary) and convert each of them to a number of standard deviations away from the
mean. To do so, subtract the mean. Then divide by the standard deviation.
Lower boundary: 1546 – 1630 = –84
–84 ÷ 84 = –1

So the lower boundary is –1 standard deviation (that is, 1 standard deviation less than the mean).
Upper boundary: 1756 – 1630 = 126
126 ÷ 84 = 1.5

So the upper boundary is 1.5 standard deviations above the mean.
You need to find the probability that a random variable distributed according to the standard normal
distribution falls between –1 and 1.5.

Use the approximate areas under the normal curve. Approximately 34 + 34 = 68% falls within 1
standard deviation above or below the mean, so 68% accounts for the –1 to 1 portion of the standard
deviation. What about the portion from 1 to 1.5?

Approximately 14% of the bolts fall between 1 and 2 standard deviations above the mean. You are
not expected to know the exact area between 1 and 1.5; however, since a normal distribution has its
hump around 0, more than half of the area between 1 and 2 must fall closer to 0 (between 1 and 1.5).
So the area under the normal curve between 1 and 1.5 must be greater than half of the area, or greater
than 7%, but less than the full area, 14%.

Put it all together. The area under the normal curve between –1 and 1.5 is approximately 68% +
(something between 7% and 14%). The lower estimate is 68% + 7% = 75% and the upper estimate is
68% + 14% = 82%.
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Re: The length of bolts made in factory Z is normally distribute [#permalink]
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Re: The length of bolts made in factory Z is normally distribute [#permalink]
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