Griffiths D. Head First Statistics

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Probability tables allow you to look up the probability P(Z < z) where

z is some value. The problem is you don’t always want to find this

sort of probability; sometimes you want to find the probability that a

continuous random variable is greater than z, or between two values.

How can you use probability tables to find the probability you need?

The big trick is to find a way of using the probability tables to get to

what you want, usually by finding a whole area and then subtracting

what you don’t need.

Probability tables give

us this probability.

Finding P(Z > z)

We can find probabilities of the form P(Z > z) using

In other words, take the area where Z < z away from the total probability.

Finding P(a < Z < b)

Finding this sort of probability is slightly more complicated to calculate,

but it’s still possible. You can calculate this sort of probability using

In other words, calculate P(Z < b), and take away the area for P(Z < a).

–

P(a < Z < b)

P(Z < b)

P(Z < a)

P(Z > z)

P(Z < z)

We’ve already used this to find

the probability that Julie is

taller than her date.

P(Z > z) = 1 - P(Z < z)

P(a < Z < b) = P(Z < b) - P(Z < a)

Probability Tables Up Close

You could use this to find the

probability that the height of Julie’s

date is within a particular range.

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I’ve heard of the term “Gaussian.”

What’s that?

A: Another name for the normal

distribution is the Gaussian distribution. If

you hear someone talking about a Gaussian

distribution, they’re talking about the same

thing as the normal distribution.

Are all normal probability tables the

same?

A: All normal probability tables give the

same probabilities for your values. However,

there’s some variation between tables as to

what’s actually covered by them.

Variation? What do you mean?

A: Some tables and exam boards use

different degrees of accuracy in their

probability tables. Also, some show the

tables in a slightly different format, but still

give the same information.

So what should I do if I’m taking a

statistics exam?

A: First of all, check what format of

probability table will be available to you while

you’re sitting the exam. Then, see if you can

get a copy.

Once you have a copy of the probability

tables used by your exam board, spend time

getting used to using them. That way you’ll

be off to a flying start when the exam comes

around.

Finding the probability of a range

looks kinda tricky. How do I do it?

A: The big thing here is to think about

how you can get the area you want using

the probability tables. Probability tables

generally only give probabilities in the form

P(Z < z) where z is some value. The big trick,

then, is to rewrite your probability only in

these terms.

If you’re dealing with a probability in the form

P(a < Z < b)—that is, some sort of range—

you’ll have two probabilities to look up, one

for P(Z < a) and the other for P(Z < b). Once

you have these probabilities, subtract the

smallest from the largest.

Do continuous distributions have

a mode? Can you find the mode of the

normal distribution?

A: Yes. The mode of a continuous

probability distribution is the value where

the probability density is highest. If you draw

the probability density, it’s the value of the

highest point of the curve.

If you look at the curve of the normal

distribution, the highest point is in the middle.

The mode of the normal distribution is μ.

What about the median?

A: The median of a continuous probability

distribution is the value a where

P(X < a) = 0.5. In other words, it’s the value

that area of the probability density curve in

half.

For the normal distribution, the median is

also μ. The median and mode don’t get used

much when we’re dealing with continuous

probability distributions. Expectation and

variance are more important.

What’s a standard score?

A: The standard score of a variable is

what you get if you subtract its mean and

divide by its standard deviation. It’s a way

of standardizing normal distributions so

that they are transformed into a N(0, 1)

distribution, and that gives you a way of

comparing them. Standard scores are

useful when you’re dealing with the normal

distribution because it means you can look

up the probability of a range using standard

normal probability tables.

The standard score of a particular value also

describes how many standard deviations

away from the mean the value is, which

gives you an idea of its relative proximity to

the mean.

no dumb questions

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It’s time to put your probability table skills to the test. See if you

can solve the following probability problems.

1. P(Z < 1.42)

2. P(-0.15 < Z < 0.5)

3. P(Z > z) = 0.1423. What’s z?

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It’s time to put your probability table skills to the challenge. See

if you can solve the following probability problems.

1. P(Z < 1.42)

We can find this probability by looking up 1.42 in the

probability tables. This gives us

P(Z < 1.42) = 0.9222

2. P(-0.15 < Z < 0.5)

For this one, look up P(Z < 0.5), and subtract P(Z < -0.15)

P(-0.15 < Z < 0.5 = P(Z < 0.5) - P(Z < -0.15)

= 0.6915 - 0.4404

= 0.2511

3. P(Z > z) = 0.1423. What’s z?

P(Z > z) = 0.1423

This is a slightly different problem. We’re given the

probability, and need to find the value of z.

We know that P(Z > z) = 0.1423, which means that

P(Z < z) = 1 - 0.1423

= 0.8577

he next thing to do is find which value of z has

a probability of 0.8577. Looking this up in the

probability tables gives us

z = 1.07

P(Z > 1.07) = 0.1423

1.42

0.5

-.015

sharpen solution

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Julie has a problem. When we calculated the probability of her date

being taller than her, we failed to take her high heels into account.

See if you can find the probability of Julie’s date being taller than

her while she’s wearing shoes with 5 inch heels.

As a reminder, Julie is 64 inches tall and X ~ N(71, 20.25) where X

is the height of men in Statsville.

Wait a sec, if I wear my

5-inch heels, I’m much taller.

Won’t that affect the probability

that my date is taller than me?

356 Chapter 8

Julie has a problem. When we calculated the probability of her date being taller than her, we

failed to take her high heels into account. See if you can find the probability of Julie’s date being

taller than her while she’s wearing shoes with 5 inch heels.

As a reminder, Julie is 64 inches tall and X ~ N(71, 20.25) where X is the height of men in

Statsville.

When Julie is wearing 5 inch high heels, her height is 69 inches. We need to find P(X > 69).

We need to start by finding the standard score of 178 so that we can use probability tables to look up the

probabilities.

Z = X - μ

= 69 - 71

4.5

= -2

4.5

= -0.44 (t

o 2 decima

l places)

Now we’ve found z, we need to find P(Z > z) i.e. P(Z > -0.44)

P(Z > -0.44) = 1 - P(Z < -0.44)

= 1 - 0.3300

= 0.67

So the probability that Julie’s date is taler than her when she’s wearing shoes with a 5 inch heel is 0.67.

The variance is 20.25, so the standard

deviation is the square root, 4.5.

-0.44

exercise solution

So, I can wear my highest heels,

and there’s still a 67% chance

he’ll be taller? Sweet!

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The Case of the Missing Parameters

Will at Manic Mango Games has a problem. He needs to give his

boss the mean and standard deviation of the number of minutes

people take to complete level one of their new game. This shouldn’t

be difficult, but unfortunately a ferocious terrier has eaten the piece of

paper he wrote them on.

Will only has three clues to help him.

First of all, Will knows that the number of minutes people spend

playing level one follows a normal distribution.

Secondly, he knows that the probability of a player playing for

less than 5 minutes is 0.0045.

Finally, the probability of someone taking less than 15 minutes to

complete level one is 0.9641.

How can Will find the mean and standard deviation?

Five Minute

Mystery

Five Minute

Mystery

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The Case of the Missing Parameters: Solved

How can Will find the mean and standard deviation?

Will can use probability tables and standard scores to get expressions for

the mean and standard deviation that he can then solve.

First of all, we know that P(X < 5) = 0.0045. From

probability tables, P(X < z

) where z

= -2.61, which

means that the standard score of 5 is -2.61. If we put

this into the standard score formula, we get

-2.61 = 5 - μ

Similarly, P(X < 15) = 0.9641, which means that the standard score of

15 is 1.8. This gives us

1.8 = 15 - μ

This gives us two equations we can solve to find μ and σ.

-2.61σ = 5 - μ

1.8σ = 15 - μ

If we subtract the first equation from the second, we get

1.8σ + 2.61σ = 15 - μ - 5 + μ

4.41σ = 10

σ = 2.27

If we then substitute this into the second equation, we get

1.8

2.27 = 15 - μ

μ = 15 - 4.086

= 10.914

In other w

ords,

μ = 10.914

σ = 2.27

Five Minute

Mystery

Solved

Five Minute

Mystery

Solved

This is a pair of equations

we can now solve.

These are the values of μ and σ.

five minute mystery solution

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And they all lived happily ever after

Just as the odds predicted, Julie’s latest blind date was a success! Julie had

to make sure her intended soulmate was compatible with her shoes, so she

made sure she wore her highest heels to put him to the test. What’s more,

he was already at the venue when she arrived, so she didn’t have to wait

around.

The first thing he said to me

was how much he liked my shoes.

We’re clearly made for each other.

We’re not entirely sure

whether she’s referring to

her date or her shoes, but

at least she’s happy.

But it doesn’t stop there.

Keep reading and we’ll show you more things you can do

with the normal distribution. You’ve only just scratched the

surface of what you can do.

The normal distribution forms the shape of a

symmetrical bell curve. It’s defined using N(μ, σ

To find normal probabilities, start by identifying the

probability range you need. Then find the standard score

for the limit of this range using

Z = X - μ where Z ~ N(0, 1).



You find normal probabilities by looking up your standard

score in probability tables. Probability tables give you

the probability of getting this value or lower.

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