Confidence intervals and sample size

Next, we are going to construct a 1-sample Proportion confidence interval using

𝑝 ̂ and �̂�

1 – sample Confidence Interval Estimate for Proportions. The equation will look

like this: �̂� ± 𝑍∗ (√ �̂�∗�̂�

𝑛 )

Notice that we are using a Z – critical value for this confidence interval because

we are referring to a proportion.

Everything after the ± sign is called the Margin of Error.

Margin of Error (ME) = 𝑍∗ (√ �̂�∗�̂�

𝑛 )

The Standard Error (SE) = (√ �̂�∗�̂�

𝑛 )

You will need to calculate the Z- critical values for this confidence interval. To

calculate the Z-critical value we will use the =NORM.S.INV( ) function in Excel.

For a 95% confidence interval, you will take 1 - .95 = .05. α = .05. But since this is

a confidence interval and we will need to add AND subtract from the mean we

will take .05/2 = .025. The new α = .025. But remember how Excel puts functions

in a less than form. To find the value we will use in the Excel function we will take

1 - .025 = .975. This is the alpha value you use in the Excel function.

Z-Critical Values do not have degrees of freedom so all you need to do is plug in

.975 into the function and hit Enter.

In Excel hit the = sign, then NORM.S.INV( .975), then close the parentheses, and

hit Enter.

The Z-Critical Value for a 95% confidence interval is 1.96.

Let’s look at an example. Recall from Week 3 that we calculated a p and q. p was

the number of successes below the average. From my example we see that I had

7 observations below the average where:

𝑝 ̂ = .70 and

�̂� = 30

I want to calculate a 95% confidence interval the proportion of car prices that will

fall below the average.

Using this equation, we will plug in what we know.

�̂� ± 𝑍∗ (√ �̂� ∗ �̂�

𝑛 )

. 70 ± 1.96 (√ . 70 ∗ .30

10 )

. 70 ± 1.96(. 1449137)

. 70 ± .28403

(.4159, .98403)

The 95% confidence interval is 41.6% to 98.4%

We are 95% confident that the proportion of cars sampled that will fall below the

average goes from 41.6% to 98.4%.