Griffiths D. Head First Statistics

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constructing confidence intervals

Four steps for finding confidence intervals

Here are the broad steps involved in finding confidence intervals. Don’t

worry if you don’t get what each step is about just yet, we’ll go through

them in more detail soon.

So can you construct a confidence

interval for any population statistic?

A: Broadly speaking, you can construct

a confidence interval for any population

statistic where you know what the sampling

distribution is like. We’ve looked at sampling

distributions for the mean and proportion, so

we can construct confidence intervals for

both of these.

What about the variance? Can we

construct a confidence interval for that?

A: Theoretically, yes, but we haven’t

covered the sampling distribution for the

variance, and we’re not going to. It’s more

common to construct confidence intervals

for the mean and proportion, and these are

what tend to be covered by statistics exams.

Do these steps relate to the

confidence interval for the mean or the

confidence interval for the proportion?

A: They’re general steps that apply to

either. You can use them for the population

mean and for the population proportion.

Does it matter how the population

is distributed?

A: The key thing is the sampling

distribution of the statistic you’re trying to

construct a confidence interval for. If you

want a confidence interval for the mean, you

need to know the sampling distribution of

means, and if you want a confidence interval

for the proportion, you need to know the

sampling distribution of proportions.

The main impact the population distribution

has on the confidence interval is the effect it

has on the sampling distribution. We’ll see

how later on.

This is the population statistic

you want to construct a

confidence interval for.

Let’s see if we can construct a confidence interval for the Mighty Gumball

CEO that he can use in his television commercials. Let’s find a confidence

interval for the mean amount of time gumball flavor lasts for.

You encountered

sampling distributions in

the last chapter.

To find the confidence limits, we

need to know the level of confidence

and the sampling distribution.

Choose your population statistic

Find its sampling distribution

Decide on the level of confidence

Find the confidence limits

The probability that your interval

contains the statistic

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Step 1: Choose your population statistic

The first step is to pick the statistic you want to construct a confidence

interval for. This all depends on the problem you want to solve.

We want to construct a confidence interval for the mean amount of

time that gumball flavor lasts for, so in this case, we want to construct a

confidence interval for the population mean, μ.

Now that we’ve chosen the population statistic, we can move onto the

next step.

Step 2: Find its sampling distribution

To find a confidence interval for the population mean, we need to know

what the sampling distribution is for the mean. In other words, we

need to know what the expectation and variance of X is, and also what

distribution it follows.

Let’s start with the expectation and variance. If we go back to the work

we did in the last chapter, then the sampling distribution of means has

the following expectation and variance:

E(X) = μ Var(X) = σ

In order to use this to find the confidence interval for μ, we substitute in

values for the population variance, σ

, and the sample size, n.

But what about μ? Why

don’t we substitute in a

value for μ?

We don’t substitute in a value for μ as this is what

we’re trying to find a confidence interval for.

We’re using the sampling distribution to help us find a confidence interval

for μ, so this means that we substitute in values for everything except for μ.

By substituting in the values for σ

and n, we can use the distribution of X

to help us find the confidence interval. We’ll show you how really soon.

There’s just one problem. We don’t know what the true value of σ

is. All

we have to go on is estimates based on the sample.

constructing confidence intervals: step by step

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constructing confidence intervals

Point estimators to the rescue

So what can we use as the value for σ

Even though we don’t know what the true value is for the population

variance, σ

, we can estimate its value using its point estimator. Rather

than use σ

, we can use

in its place, or s

This means that the expectation and variance for the sampling

distribution of means is

E(X) = μ Var(X) = s

Mighty Gumball used a sample of 100 gumballs to come up with their

estimates, and they have calculated that s

= 25. This means that

Var(X) = s

= 25

100

= 0.25

There’s still one thing we have left to do. Before we can find the

confidence interval for μ, we need to know exactly how X is distributed.

This is the point estimator for the variance. We don’t know

what the true value of the population variance is, so we use the

sample variance to estimate it instead..

Assume that X ~ N(μ, σ

) and that the number in the sample is

large. What distribution does X follow? Use E(X) and Var(X) above

to help you.

494 Chapter 12

Assume that X ~ N(μ, σ

) and that the number in the sample is

large. What distribution does X follow? Use E(X) and Var(X) above

to help you.

If X follows a normal distribution, then X does too. Substituting in the point estimator for σ

, we get:

X ~ N(μ, s

/n)

or:

X ~ N(μ, 0.25)

We’ve found the distribution for X

Now that we know how X is distributed, we have enough information to

move onto the next step.

Step 3: Decide on the level of confidence

The level of confidence lets you say how sure you want to be that the

confidence interval contains your population statistic. As an example,

suppose we want a confidence level of 95% for the population mean.

This means that the probability of the population mean being inside the

confidence interval is 0.95.

The confidence level is the probability

of the population mean being inside the

confidence interval. For a confidence level

of 95%, the probability here is 0.95.

How do you think the level of confidence affects the size of the confidence

interval?

μ = x

sharpen solution

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constructing confidence intervals

So who decides what the level of confidence should be? What’s the right

level of confidence?

The answer to this really depends on your situation and how confident

you need to be that your interval contains the population statistic. A 95%

confidence level is common, but sometimes you might want a different

one, such as 90% or 99%. As an example, the Mighty Gumball CEO

might want to have a higher degree of confidence that the population

mean falls inside the confidence interval, as he intends to use it in his

television advertisements.

The key thing to remember is that the higher the confidence level is,

the wider the interval becomes, and the more chance there is of the

confidence interval containing the population statistic.

Well, why don’t we just make the

confidence interval really wide?

That way we’re bound to include

the population statistic.

The trouble with making the confidence interval

too wide is that it can lose meaning.

As an extreme example, we could say that that the mean duration of

gumball flavor is between 0 minutes and 3 days. While this is true, it

doesn’t give you an idea how long gumball flavor really lasts for. You

don’t know whether it lasts for seconds, minutes, or hours.

The key thing is to make the interval as narrow as possible, but

wide enough so you can be reasonably sure the true mean is in the

interval.

Let’s use a 95% confidence level for Mighty Gumball. That way,

there’ll be a high probability that it contains the population mean.

Now that we have the confidence level, we can move onto the final step:

finding the confidence limits.

How to select an appropriate confidence level

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Step 4: Find the confidence limits

The final step is to find a and b, the limits of the confidence interval,

which indicate the left and right borders of the range in which there’s

a 95% probability of the mean falling. The exact value of a and b

depends on the sampling distribution we need to use and the level of

confidence that we need to have.

For this problem, we need to find the 95% confidence level for the mean

duration of gumball flavor. Meaning, there must be a .95 probability

that μ lies between the a and b that we find. We also know that X

follows a normal distribution, where X ~ N(μ, 0.25).

Here’s a sketch of what we need:

We want to choose a and b so that this

area represents a probability of 0.95.

These two areas need to add up to a

probability of 0.05, so that’s 0.025

for each tail.

We can find the values of a and b using the distribution of X. In other

words, we can use the distribution of X ~ N(μ, 0.25) to find a and b, such

that P(X < a) = 0.025 and P(X > b) = 0.025.

So does that mean we use

the normal distribution to

find the confidence interval

for μ?

As X follows a normal distribution, we can use the

normal distribution to find the confidence interval.

We can do this in a similar way to how we’ve solved other problems in

the past. We calculate a standard score, and we use standard normal

probability tables to help us get the result we need.

μ = x

confidence limits

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constructing confidence intervals

We need to find z

and z

, such that P(z

< Z < z

) = 0.95.

We need to find z

and z

where P(z

< Z < z

) = 0.95. In other words,

the standardized confidence limits are given by z

and z

where

P(Z < z

) = 0.025 and P(Z > z

) = 0.025. We can find the values of z

and z

using probability tables.

2. Use probability tables to find the value of z

where P(Z > z

) = 0.025.

1. Use probability tables to find the value of z

where P(Z < z

) = 0.025.

Start by finding Z

Before we can use normal probability tables, we need to standardize X.

We know that X ~ N(μ, 0.25), so this means that if we standardize, we get

Z = X - μ where Z ~ N(0, 1)

√0.25

Here’s a sketch of the standardized version of the confidence interval.

0.025

0.95

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We need to find z

and z

such that P(z

< Z < z

) = 0.95.

If we look up the probability 0.025 in standard normal probability tables, this gives us z

= -1.96

To find z

, we need to look up a value of 0.975. This gives us z

= 1.96.

Rewrite the inequality in terms of µ

So far, we’ve found a standardized version of the confidence interval.

We’ve found that P(-1.96 < Z < 1.96) = 0.95. In other words,

(

-1.96 <

X - μ

< 1.96

)

= 0.95

0.5

2. Use probability tables to find the value of z

where P(Z > z

) = 0.025.

1. Use probability tables to find the value of z

where P(Z < z

) = 0.025.

But don’t we need a

confidence interval

for μ? How do find it?

We can find the confidence interval for μ by

rewriting the inequality in terms of μ.

If we can rewrite

-1.96 <

X - μ

< 1.96

0.5

in the form

a < μ < b

We’ll have our confidence limits for μ.

sharpen solution

This will give us

a confidence

interval for μ.

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constructing confidence intervals

Pool Puzzle

Your job is to rewrite

-1.96 < (X - μ)/0.5 < 1.96 and come

up with a confidence interval for

μ. Take snippets from the pool

and place them into the blank

lines. You may not use the same

snippet more than once.

Note: Each thing from

the pool can only be

used once!

-1.96 <

X - μ

< 1.96

0.5

-1.96 <

X - μ

0.5

-1.96

< X - μ

+ μ < X

μ <

X - μ

< 1.96

0.5

X - μ <

0.5

X < + μ

< μ

0.5

-0.98

X + 0.98

1.96

0.98

X - 0.98

X - 0.98 < μ < X + 0.98

This gives you the lefthand

side of the inequality

This gives you the righthand side

This is what you get if you

put the two sides of the

inequality together again

500 Chapter 12

Pool Puzzle Solution

Your job is to rewrite

-1.96 < (X - μ)/0.5 < 1.96 and come

up with a confidence interval for

μ. Take snippets from the pool

and place them into the blank

lines. You may not use the same

snippet more than once.

Note: Each thing from

the pool can only be

used once!

-1.96 <

X - μ

< 1.96

0.5

-1.96 <

X - μ

0.5

-1.96

< X - μ

+ μ < X

μ <

X - μ

< 1.96

0.5

X - μ <

0.5

X < + μ

< μ

0.5

-0.98

X + 0.98

1.96

0.98

X - 0.98

X - 0.98 < μ < X + 0.98

This gives you the left

hand side of the inequality.

This gives you the right hand side.

pool puzzle solution