Question

The heights of women aged 20 to 29 are approximately normal with mean 64.3 inches and standard deviation 2.7 inches. Men the same age have mean height 69.9 inches with standard deviation 3.1 inches. What is the z-score for a man 6 feet tall?

Answer 1

@SithsAndGiggles

Answer 2

what is z-score ?

Answer 3

thats what i need to find out because the question does not say the z-score its asking for the z-score a 6 feet tall man. Im confused sorry

Answer 4

\(\color{blue}{\text{Originally Posted by}}\) @SithsAndGiggles

http://www.pavementinteractive.org/wp-content/uploads/2007/08/Normal_table.gif

@Loser66, we're given a data set involving a random variable (height), which I'll call \(X\), that is normally distributed (think of the standard bell curve) with a given mean \(\mu=64.3\) and standard deviation \(\sigma=2.7\). (This is written \(X\sim \text{Norm}(\mu,\sigma)\) in some textbooks; read it as "\(X\) is normally distributed with mean \(\mu\) and std. dev. \(\sigma\).)

The z-score has to do with transforming this given random variable into a more suitable one. In particular, what we're looking for is the probability of an event of interest. For example, let's say we want to find the probability that a person chosen at random from the sample (any one woman aged 20 to 29) has a height is between two heights. This probability would be expressed as \(P(a<X<b)\) for some heights \(a,b\).

This probability is calculated by finding the area under a curve. But this is no ordinary curve, by which I mean it's not easy to just compute an integral. The curve we're using is the one described here: http://en.wikipedia.org/wiki/Normal_distribution

The transformation I mentioned is as follows: \(Z=\dfrac{X-\mu}{\sigma}\). \(Z\) follows what is called the standard normal distribution, which has \(\mu=0\) and \(\sigma=1\) (see the red curve in the first figure on the wiki page).

The z-chart I linked to is the computed area under the curve to the left of a particular \(z\) value.
\(\color{blue}{\text{End of Quote}}\)

Answer 5

Sorry about that...

Answer 6

Woaaah!! It's new to me. Thanks @SithsAndGiggles

Answer 7

@maryr13, the sample of men has \(\mu=69.9\) and \(\sigma=3.1\). Using the transformation I mentioned, the z-score for a man that is 6 ft tall, or 72 inches tall, is
\[z=\frac{72-\mu}{\sigma}=\frac{72-69.9}{3.1}=0.677\]
Now, I'm not sure if by z-score, you mean this value of \(z\), or if you mean the probability that a man is 72 in tall. If it's the latter, then refer to the table I posted (the very first link).

Answer 8

@SithsAndGiggles thank you for the help i seem to understand now :) and on the question it only mentions what is the z-score.

Answer 9

one more question: why does the problem indicate to women's height which does nothing for the question of the height of the man?

Answer 10

A trick? maybe there are other parts to the question.

Also, apparently the first table I posted is incomplete. Here's a better one, with the value you're looking for:

Answer 11

The z-score is the number of standard deviations the value is from the mean, and in which direction (larger or smaller).