The heights of males in a certain country are normally distributed with a mean of 70.2 inches and a standard deviation of 4.2 inches. What is the probability that a randomly chosen male is under the height of 65 inches?

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The heights of males in a certain country are normally distributed with a mean of 70.2 inches and a standard deviation of 4.2 inches. What is the probability that a randomly chosen male is under the height of 65 inches?

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Answer 1 · 2017-02-14T06:17:30

Since 65 is (approx.) -1.24 standard deviations from the mean of 70.2,

$P (X < 65) \approx P (Z < - 1.24)$ $P(X < 65)" "~~" "P(Z < "–"1.24)$
$P (X < 65) = 0.1075$ $color(white)(P(X < 65)" ")=" "0.1075$ .

Explanation:

If $X$ $X$ is $N (μ, σ^{2})$ $N(mu, sigma^2)$ and $Z$ $Z$ is $N (0, 1)$ $N(0,1)$ , then

$P (X < x) = P (Z < \frac{x - μ}{σ}) .$ $P(X < x)= P(Z< (x-mu)/sigma).$

This means we can find the probability $P (X < x)$ $P(X < x)$ by computing the corresponding probability from the standard normal distribution $Z$ $Z$ .

Let $X$ $X$ be the height of a random male from the given country. Then $X ~ N (70.2, 4.2)$ $X" "~" "N(70.2,4.2)$ , and so

$P (X < 65) = P (Z < \frac{65 - 70.2}{4.2})$ $P(X < 65)=P(Z < (65-70.2)/4.2)$
$P (X < 65) \approx P (Z < - 1.24)$ $color(white)(P(X < 65)) ~~ P(Z < "–"1.24)$

What this means is that the point in the $X$ $X$ distribution where $X = 65$ $X=65$ corresponds (approximately) to the point in the $Z$ $Z$ distribution where $Z = - 1.24$ $Z="–"1.24$ . In other words, 65 is (approx.) 1.24 standard deviations below the mean of 70.2.

Why bother "transforming" $X$ $X$ like this? Because it would be impossible to print a look-up table of $x$ $x$ -scores for every possible normal distribution, but we can print one table as a base, then show how to relate any normal random variable to this base one.

The table we so happen to make is for $Z$ $Z$ , the standard normal distribution. Thus, looking up $P (Z < - 1.24)$ $P(Z < "–"1.24)$ in our $z$ $z$ -score table is identical to looking up $P (X < 65)$ $P(X < 65)$ in its table (if it existed).

From the table, $P (Z < - 1.24) = 0.1075$ $P(Z < "–"1.24)=0.1075$ , and so

$P (X < 65) \approx 0.1075$ $P(X < 65) ~~ 0.1075$ .

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The heights of males in a certain country are normally distributed with a mean of 70.2 inches and a standard deviation of 4.2 inches. What is the probability that a randomly chosen male is under the height of 65 inches?

1 Answer

Explanation:

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