How Do You Use a Z-Score Table for Probability Calculations?

[tsukuba]

Homework Statement

Let X be a normal random variable with mean 80 and standard deviation 5. What is the probability X is greater than 87? Less than 74?
I understand how to compute the z score but I'm confused as to when I have to keep the number from the z-table or subtract 1

Homework Equations

z= x-μ /σ

The Attempt at a Solution

z = 1.4
P (z > 87 ) = ??
I found the z score to be 0.9192 but do I keep this value as my answer or do I do 1 - 0.9192z=-1.2
P (z < 74)= ??
again, I found z score to be .1151
So I keep this value or 1 - 0.1151 and why?
Thank you

 

[Mark44]

Homework Statement

Let X be a normal random variable with mean 80 and standard deviation 5. What is the probability X is greater than 87? Less than 74?
I understand how to compute the z score but I'm confused as to when I have to keep the number from the z-table or subtract 1

Homework Equations

z= x-μ /σ

You need parentheses when you write things like this. It should be z = (x - μ )/σ. As you wrote it, it would be interpreted as z = x - (μ /σ).

The Attempt at a Solution

z = 1.4
P (z > 87 ) = ??

No, the above should be P(X > 87). You get the z-score by transforming the inequality, like so:
P(X > 87) = $P(\frac{X - 80}{5} > \frac{87 - 80}{5}) = P(z > \frac{7}{5}) = P(z > 1.4)$

I found the z score to be 0.9192 but do I keep this value as my answer or do I do 1 - 0.9192

No, the z-score is 1.4, and the probability associated with it is 0.9192. This probability is the area under the standard normal curve (the "bell curve") between -∞ and 1.4.

It's very helpful to draw a quick sketch of the bell curve, including your z-score and the probability value you get from the table. With a sketch you can easily see which area under the curve you're interested in.

z=-1.2
P (z < 74)= ??
again, I found z score to be .1151
So I keep this value or 1 - 0.1151 and why?
Thank you

 

[Ray Vickson]

Homework Statement

Let X be a normal random variable with mean 80 and standard deviation 5. What is the probability X is greater than 87? Less than 74?
I understand how to compute the z score but I'm confused as to when I have to keep the number from the z-table or subtract 1

Homework Equations

z= x-μ /σ

The Attempt at a Solution

z = 1.4
P (z > 87 ) = ??
I found the z score to be 0.9192 but do I keep this value as my answer or do I do 1 - 0.9192z=-1.2
P (z < 74)= ??
again, I found z score to be .1151
So I keep this value or 1 - 0.1151 and why?
Thank you

As Mark44 has said: draw a picture---even a very crude (but symmetric) picture will do. For any z, P(Z<z) is the area to the left of the point z under the graph of the unit normal density, so if z < 0, P(Z < z) < P(Z < 0) = 1/2, while P(Z > z) > P(Z > 0) = 1/2. So, if you are given a probability value P(Z < z ) < 1/2 or P(Z > z) > 1/2, you know that z < 0. If you are given the opposites of these, you know that z > 0.

 

[tsukuba]

Thank you both
Did well on my exam