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coldgirl [10]
2 years ago
12

Consider a normal distribution curve where the middle 85 % of the area under the curve lies above the interval ( 8 , 14 ). Use t

his information to find the mean, \mu , and the standard deviation, \sigma , of the distribution.
Mathematics
1 answer:
NeTakaya2 years ago
7 0

Answer:

\mu = 11

\sigma = 2.08

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Middle 85%.

Values of X when Z has a pvalue of 0.5 - 0.85/2 = 0.075 to 0.5 + 0.85/2 = 0.925

Above the interval (8,14)

This means that when Z has a pvalue of 0.075, X = 8. So when Z = -1.44, X = 8. So

Z = \frac{X - \mu}{\sigma}

-1.44 = \frac{8 - \mu}{\sigma}

8 - \mu = -1.44\sigma

\mu = 8 + 1.44\sigma

Also, when X = 14, Z has a pvalue of 0.925, so when X = 8, Z = 1.44

Z = \frac{X - \mu}{\sigma}

1.44 = \frac{14 - \mu}{\sigma}

14 - \mu = 1.44\sigma

1.44\sigma = 14 - \mu

Replacing in the first equation

\mu = 8 + 1.44\sigma

\mu = 8 + 14 - \mu

2\mu = 22

\mu = \frac{22}{2}

\mu = 11

Standard deviation:

1.44\sigma = 14 - \mu

1.44\sigma = 14 - 11

\sigma = \frac{3}{1.44}

\sigma = 2.08

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