Answered

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X is a normally distributed random variable with mean 46 and standard deviation 22.
What is the probability that X is between 2 and 90?

A. .99
B. .95
C. .85
D. .65

Sagot :

sqdancefan

Answer:

  B.  0.95

Step-by-step explanation:

The probability that the value of the random variable is in the given range is the area under the normal probability distribution curve over that range. The value of this area is conveniently provided by any number of calculators, apps, or spreadsheets. It can also be determined using the "empirical rule" that tells you the probability over certain normalized ranges.

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normalized range

A standard normal probability density function (PDF) has a mean of 0 and a standard deviation of 1. The range in this problem can be normalized to some number of standard deviations above or below the mean. The mean is given as 46, and the standard deviation is given as 22.

In this problem, the lower limit of the range is ...

  z = (2 -46)/22 = -2

standard deviations from the mean.

Similarly, the upper limit is ...

  z = (90 -46)/22 = +2

standard deviations from the mean.

empirical rule

The empirical rule tells us that 95% of the area of the PDF lies between -2 and +2 standard deviations from the mean.

  P(2 < X < 90) ≈ 0.95

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Additional comment

The empirical rule for other ranges is ...

  ± 1 standard deviation: 68%

  ±2 standard deviations: 95%

  ±3 standard deviations: 99.7%

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