Griffiths D. Head First Statistics

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We interrupt this chapter to bring you...

Hello, and welcome back to Who Wants

To Win A Swivel Chair, Statsville’s favorite

quiz show. We’ve got some more fiendishly

difficult questions on tonight’s show.

382 WHO WANTS TO WIN A SWIVEL CHAIR

We’ve got some more great questions lined

up for you today, so let’s get on with the show. In

this round I’m going to ask you forty questions, and you

need to get thirty or more right to get through to the

next round. Or you can walk away and take a consolation

prize. For each question there are four possible answers.

The title of this round is “Even More About Me.”

Good luck!

Here are the first five questions for Round Two.

The questions are all about the game show host.

D: $4,000

B: $2,000

C: $3,000

3. On average, how much does he spend on suits each month

A: $1,000

D: Bird on a Wire

B: Curse of the Were-Rabbit

C: Mousehunt

2. What is the favorite film of his cat

A: A Fish Called Wanda

D: All the President’s Men

B: The Italian Job

C: Lawrence of Arabia

1. What is his favorite film

A: The Day of The Jackal

D: Four times a month

B: Twice a month

C: Three times a month

4. How often does he have his hair cut?

A: Once a month

D: www.starbuzzcoffee.com

B: www.gregs-list.net

C: www.you-cube.net

5. What is his favorite web site

A: www.fatdanscasino.com

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How would you find the probability of getting at least 30 out of

40 questions correct? What steps would you need to go through

to get the right answer? How would you find the mean and

variance?

We’re not asking you to find the probability—just say how you’d

go about finding it.

Should we play, or walk away?

As before, it’s unlikely you’ll know the game show host well enough to

answer questions about him. It looks like you’ll need to give random

answers to the questions again.

So what’s the probability of getting 30 or more questions right out of 40?

That will help us determine whether to keep playing, or walk away.

384 WHO WANTS TO WIN A SWIVEL CHAIR

How would you find the probability of getting at least 30 out of

40 questions correct? What steps would you need to go through

to get the right answer? How would you find the mean and

variance?

We’re not asking you to find the probability—just say how you’d

go about finding it.

There are 40 questions, which means there are 40 trials. The outcome of each trial can be a success or failure,

and we want to find the probability of getting a certain number of successes. In order to do this, we need to use

the binomial distribution. We use n = 40, and as each question has four possible answers, p is 1/4 or 0.25..

If X is the number of questions we get right, then we want to find P(X > 30). This means we have to calculate

and add together the probabilities for P(X = 30) up to P(X = 40).

We can find the mean and variance using n, p and q, where q = 1 - p. The mean is equal to np, and the variance is

equal to npq. This gives us a mean of 40 x 0.25 = 10, and a variance of 40 x 0.25 x 0.75 = 7.5.

But doing all of those

calculations is going to be

horrible. Isn’t there an

easier way?

Using the binomial distribution can be a lot of work.

In order to find the probability that we answer 30 or more questions

correctly, we need to add together 11 individual probabilities. Each of

these probabilities is tricky to find, and it would be very easy to make a

mistake somewhere along the way.

What we really need is an easier way of calculating binomial

probabilities.

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Wouldn't it be dreamy if there was a

way of making other distributions as easy

to work with as the normal? But I know it's

just a fantasy…

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Normal distribution to the rescue

We’ve seen that life with the binomial distribution can be tough at times. Some of

the calculations can be tricky and repetitive, which in turn means that it’s easy to

make mistakes and spend a lot of time only to come up with the wrong answer.

Sound hopeless? Don’t worry, there’s an easy way out.

In certain circumstances, you can use the normal distribution to approximate the

binomial distribution.

You’re saying the normal

distribution can approximate

the binomial? I thought the

Poisson did that. What gives?

The Poisson distribution can approximate the binomial in

some situations, but the normal can in others.

Knowing how to approximate the binomial distribution with other distributions is

useful because it can cut down on all sorts of complexities, and in some situations

the Poisson distribution can help us work out some tricky binomial probabilities.

In certain other circumstances, we can use the normal distribution to approximate

the binomial instead. There are some huge advantages with this, as it means that

instead of performing calculations, we can use normal probability tables to simply

look up the probabilities we need.

All we need to do is figure out the circumstances under which this works.

It’s been a while since we looked at how we could use the

Poisson distribution to approximate the binomial. Under what

circumstances is it appropriate?

B(n, p) can be approximated by the Poisson when n > 50 and p < 0.1.

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BE the Distribution

Below you’ll see some binomial

distributions for different values of n

and p. Your job is to play like you’re

the distribution and say

which one you think can best

be approximated by the normal.

Take a good look at the shape of

each distribution and say which

one is most normal.

n = 5

p = 0.1

1 2

4 5

0.2

0.1

0.3

0.4

0.5

0.6

n = 5

p = 0.5

1 2

0.2

0.1

0.3

0.4

n = 20

p = 0.1

1 2 3 4

0.2

0.1

0.3

5 6 7

8 9 10

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n = 20

p = 0.5

2 4 6 8

0.2

0.1

0.3

0 14 16 18 201 19

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BE the Distribution Solution

Below you’ll see some binomial

distributions for different values of n

and p. Your job is to play like you’re

the distribution and say which

one you think can best be

approximated by the normal.

Take a good look at the shape of

each distribution and say which

one is most normal.

Out of all these distributions, this is the one that can best be

approximated by the normal distribution. When n = 20 and p = 0.5,

the shape of the distribution is very similar to that of the normal.

This distribution is symmetrical, but it’s

difficult to pick out a smooth bell-shaped

curve. It’s unlikely that using the normal

distribution to approximate this would be

accurate enough.

n = 5

p = 0.1

1 2

4 5

0.2

0.1

0.3

0.4

0.5

0.6

n = 20

p = 0.1

1 2 3 4

0.2

0.1

0.3

0 5 6 7 8 9 10

n = 5

p = 0.5

1 2

0.2

0.1

0.3

0.4

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n = 20

p = 0.5

2 4 6 8

0.2

0.1

0.3

0 14 16 18 201 19

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Under certain circumstances, the shape of the binomial distribution looks

very similar to the normal distribution. In these situations, we can use the

normal distribution in place of the binomial to give a close approximation of its

probabilities. Instead of calculating lots of individual probabilities, we can look up

whole ranges in standard normal probability tables.

So under what circumstances can we do this?

We saw in the last exercise that the binomial distribution looks very similar to the

normal distribution where p is around 0.5, and n is around 20. As a general rule,

you can use the normal distribution to approximate the binomial when np and nq

are both greater than 5.

Finding the mean and variance

Before we can use normal probability tables to look up probabilities, we need to know what the

mean and variance is so that we can calculate the standard score. We can take these directly from

the binomial distribution. When we originally looked at the binomial distribution, we found that:

μ = np and σ

= npq

We can use these as parameters for our normal approximation.

Approximating the Binomial Distribution

If X ~ B(n, p) and np > 5 and nq > 5, you can use X ~ N (np, npq) to

approximate it.

Vital StatisticsVital Statistics

When to approximate the binomial distribution with the normal

Some text

books use a

criteria of

np > 10 and

nq > 10.

If you’re taking a statistics

exam, make sure you

check the criteria used by

your exam board.

n is the number of values,

p is the probability of

success, and q is 1 - p.

We can use the normal distribution to

approximate binomial probabilities when np

and nq are both greater than 5. These

values of n, p, and q give us a nice, smooth

shape that’s pretty close to the normal.

npq

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Before we use the normal distribution for the full 40 questions for Who Wants To Win A Swivel

Chair, let’s tackle a simpler problem to make sure it works. Let’s try finding the probability that

we get 5 or fewer questions correct out of 12, where there are only two possible choices for

each question.

Let’s start off by working this out using the binomial distribution. Use the binomial distribution

to find P(X < 6) where X ~ B(12, 0.5).