Online Assignment #3 Confidence Intervals for the ProportionNow we’re going to estimate a different parameter (number describing a proportion) by looking at a sample, and it’s a parameter we haven’t mentioned yet.

We call it the population proportion, and its symbol is p. Notice that it’s not a Greek letter like some other parameters, µ and ơ. That’s because the Greek P is busy being the ratio between the circumference of a circle and its radius (in other words, π.

Say we want to estimate the proportion of Mendocino College students who are teenagers. What better place to start than with our data base? If you Sort(A) List 2, you’ll see that it contains 40 teenagers (the last 19 is L2 (40 ).

We give the 40 the symbol x. It’s the number of members of the sample who have the characteristic we’re looking at (being a teenager, in this case). We already have a symbol and name for the 81 – it’s n, the sample size.

40 out of 81 gives us the fraction 4181 , using the formula x

n . And there’s a delightful

symbol for this formula: p̂. Best of all, it’s pronounced p-hat. It’s the sample proportion, and it’s a statistic because it describes a sample.

So, same story as with µ, we start by using p̂ as a point estimate for p, and it’s easier if you make it a decimal:

p̂= xn=40

81≈ 0.494

But again, the same story as with µ: what if we used some other sample of 81 Mendocino College students? It’s unlikely there would be exactly 40 teenagers. So we make a confidence interval for the proportion, and it looks like this:

p̂−E<μ< p̂+E

The way E is calculated is different from how it was calculated for estimating the mean, but the calculator knows the difference.

Here’s the thing: when you have a quantitative variable like height (see OA #2) you get a data set with numbers, and the set has a mean and standard deviation. But when you have a qualitative variable (is a teenager or isn’t a teenager), there’s no list of numbers, no mean, and no standard deviation, and all you have is the

fraction that are teenagers (which for some reason is called a proportion in statistics). And that’s true even if the variable is multinomial.

Here’s how you do it on the calculator. Go to Stat Tests. You won’t see the option you want at first:

So scroll down to Option A: 1-PropZInt:

When you select 1-PropZInt, your screen will look like this, though there may be different numbers or none for x and n.

Put in 40 for x and 81 for n, put in the confidence level you want if it isn’t there already, and go to calculate:

Just like with estimating the mean, the calculator expresses the answer in interval notation, so change it to the double-less-than notation, only this time the parameter p goes in the middle. Also, round to the nearest thousandth (tenth of a percent):

0.351< p<0.637

or

35.1 %< p<63.7 %

I’d pronounce this, “p is between 35.1% and 63.7%

So here’s the assignment:

1) Do Worksheet #20 on p. 136. You don’t need to submit it, but if you have questions, let me know.

2) Find and write the 99%, the 95%,

the 90%

confidence interval for

a) the proportion of Mendocino College students who are natives of Lake or Mendocino County

b) the proportion of Mendocino College students who pick green or a color with the word “green” in it as their favorite color

c) the proportion of Mendocino College students who have no pets

Round answers to the nearest thousandth, or tenth of a percent.

3) Follow the explanation in Online Assignment #2, Problem #3 of finding the margin of error and width of confidence intervals, and state the margins of error and the widths of the nine intervals you found in Problem #2.

Compare the widths of confidence intervals at different confidence levels. What do you notice as the level of confidence decreases?

4) Now the poll.

a) Find and write the 99%

the 95%

and the 90%

confidence intervals for the proportion of people polled who have lost their jobs due to the pandemic (round to the nearest thousandth, or tenth of a percent).

b) Find and write the 99%

the 95%

and the 90%

confidence intervals for the proportion of people polled whose normal routines have changed a lot due to the pandemic (round to the nearest thousandth, or tenth of a percent).

c) Figure out the average number of people sheltering in place. Round to the nearest tenth. Hint: If 4 people are sheltering with 1 person, that’s 4 people altogether (4 times 1), if 8 people are sheltering with 2 people, that’s 16 altogether (8 times 2), and so on. Find the total number of people sheltering and divide by the number of people who responded, which you can tell by adding up the frequencies.