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29 Sep 2024, 03:37
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
44% (00:45) correct
55% (00:56) wrong based on 9 sessions
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The figure above shows the standard normal distribution, with mean 0 and standard deviation 1, including approximate percents of the distribution corresponding to the six regions shown.
The random variable Y is normally distributed with a mean of 470, and the value Y = 340 is at the 15th percentile of the distribution. Of the following, which is the best estimate of the standard deviation of the distribution?
A. 125
B. 135
C. 145
D. 155
E. 165
Attachment:
1234.jpg [ 27.19 KiB | Viewed 2286 times ]
29 Jan 2025, 12:12
Kudos
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OE
Since you know that the distribution of the random variable Y is normal with a mean of 470 and that the value 340 is at the 15th percentile of the distribution, you can estimate the standard deviation of the distribution of Y using the standard normal distribution. You can do this because the percent distributions of all normal distributions are the same in the following respect: Te percentiles of every normal distribution are related to its standard deviation in exactly the same way as the percentiles of the standard normal distribution are related to its standard deviation. For example, approximately 14% of every normal distribution is between 1 and 2 standard deviations above the mean, just as the fgure above illustrates for the standard normal distribution.
From the fgure, approximately 2% + 14%, or 16%, of the standard normal distribution is less than -1. Since 15% < 16%, the 15th percentile of the distribution is at a value slightly below -1. For the standard normal distribution, the value -1 represents 1 standard deviation below the mean of 0. You can conclude that the 15th percentile of every normal distribution is at a value slightly below 1 standard deviation below the mean.
For the normal distribution of Y, the 15th percentile is 340, which is slightly below 1 standard deviation below the mean of 470. Consequently, the difference 470 - 340, or 130, is a little greater than 1 standard deviation of Y; that is, the standard deviation of Y is a little less than 130. Of the answer choices given, the best estimate is 125, since it is close to, but a little less than, 130.
The correct answer is Choice A.
Since you know that the distribution of the random variable Y is normal with a mean of 470 and that the value 340 is at the 15th percentile of the distribution, you can estimate the standard deviation of the distribution of Y using the standard normal distribution. You can do this because the percent distributions of all normal distributions are the same in the following respect: Te percentiles of every normal distribution are related to its standard deviation in exactly the same way as the percentiles of the standard normal distribution are related to its standard deviation. For example, approximately 14% of every normal distribution is between 1 and 2 standard deviations above the mean, just as the fgure above illustrates for the standard normal distribution.
From the fgure, approximately 2% + 14%, or 16%, of the standard normal distribution is less than -1. Since 15% < 16%, the 15th percentile of the distribution is at a value slightly below -1. For the standard normal distribution, the value -1 represents 1 standard deviation below the mean of 0. You can conclude that the 15th percentile of every normal distribution is at a value slightly below 1 standard deviation below the mean.
For the normal distribution of Y, the 15th percentile is 340, which is slightly below 1 standard deviation below the mean of 470. Consequently, the difference 470 - 340, or 130, is a little greater than 1 standard deviation of Y; that is, the standard deviation of Y is a little less than 130. Of the answer choices given, the best estimate is 125, since it is close to, but a little less than, 130.
The correct answer is Choice A.
