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Calculating Standard Error of a Sample Proportion

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In this lesson, I'm going to show you how to calculate standard error for sample means, for sample proportions when this population standard deviation is unknown, and for sample proportions when the population standard deviation is known. And I'm going to give you an example of each, so you can also practice identifying when to use, which of the three formulas to calculate standard error. The first example we're going to look at is we're asked to calculate the standard error for the amount of fallen snow, in inches, that is recorded for one week in Minneapolis. So the very first question you always want to ask yourself is what type of data am I dealing with? Is this quantitative or is it qualitative, which is also known as categorical data?

Well, inches of snow, that is a quantitative variable, so we're looking for the standard error of a sample mean. So in that case, our formula is going to be s over square root of n. S stands for the standard deviation of this sample, and n is your sample size, which in this case, we have seven pieces of data for the weeks, so our sample size is seven.

So I'm going to show you how to do this in the calculator. And what we want to do is we're going to enter our data into a list. So to get there, go ahead and hit your Stat button, and hit Enter for edit, and we're going to insert our data into list one.

So first, we have 1.5 inches of snow, so 1.5 Enter, then 3 Enter. Then it snowed a lot, 4.75 inches up to 8 inches. Then it tapered off to 0.3 inches. Then on Friday, they got 2 inches of snow. And finally on Saturday, 2.95 inches of snow.

And I'm going to exit out of this screen once I've entered all of my data. And I do that by hitting 2nd Mode. Now, to get s, the standard deviation of this sample, I need to get the sample statistics from that list of data. And to do so, again, hit your Stat button, scroll over to Calc, and we're interested in this first function, one variant statistics.

Go ahead and hit Enter. And we want it for list one, so you're going to hit 2nd 1, and you can see the L1 in the upper left hand corner above the one button. Hit Enter, and we get all sorts of useful data for this set of data.

So we have the x bar, which is the mean. We have s of x. We have sigma of x. We have the sample size, which is seven.

You also get your five number summary, which is really useful for other types of problems. But for this problem, we are interested, like I said, in the standard deviation of the sample. Now, this is a sample. It is not a population, so we want to use the value for s of x, not sigma of x. Remember, population statistics are noted with Greek letters.

But in this case, like I said, it's just a sample, so we're going to use the 2.526 is what I'll round that too. So 2.526 divided by the square root of 7. That's my sample size. And when I calculate that, I get a standard error of 0.955.

Next, I'm going to show you how to calculate the standard error of the sample in Excel. Now, I'm going to show you how to use Excel to calculate the standard error of the sample mean. The very first thing we need to do is to calculate s, which is the standard deviation from your sample.

So I'm going to go to my Formulas tab. I'm going to insert the formula. And I'm going to go under Statistics and look for the standard deviation of a sample. which is indicated here with the dot s. The dot p is the standard deviation of the population, but remember, we're just dealing with a sample.

So I'm going to hit Enter, and I'm going to insert my data, separating each data value with a comma. So my first data value was 1.5 inches of comma 3 comma 4.75. And then I had 8 inches of snow, 0.3 inches of snow, 2 inches of snow, and finally 2.95.

I'm going to hit Enter. And notice how I get the same value that we got in our calculator under s of x. Now, I'm going to finish calculating the standard mean, which was s divided by the square root of n. And I'm going to go ahead and the first you have to do is hit the equal sign, and my s value is in cell A1.

So you just have to click on A1 with your mouse, and that automatically inserts it. Divided by the square root of n. To get the square root, again, you have to insert a formula, but the square root is just under math and trigonometry, and it's indicated with an SQRT. So hit Enter, and our sample size was 7, hit Enter, and notice how we get the same value we did in our calculator for standard error, 0.955.

In this example, a survey was conducted at the local high schools to find out about underage drinking. Of the 523 students who replied to the survey, 188 had replied, yes, that they have drank some amount of alcohol. And we're asked to calculate the standard error for the sample proportion.

Now, these students are either answering yes or no on the survey. Yes, I've drank some amount of alcohol or no I have not. That is qualitative data, also known as categorical data. Therefore, we're dealing with a sample proportion.

Now, whenever you're dealing with a sample proportion, the next question you need to ask yourself is do I know the population standard deviation? Well in this case, I don't have any of that information, so therefore, the formula I need to use to calculate the standard error is p hat q hat divided by n all under the square root. We're actually going to use the data that was given to us, which are estimates-- that's what the hat indicates-- in order to calculate the standard error.

So the first thing I need to do is I need to figure out what p hat is, based off of the information given to me. So p hat in this case is what we're interested in, and that is how many have answered yes to participating in underage drinking. So that would be 188 out of 523 students to get me about 36% of the students.

Now, I also need the complement, which would be q hat. Sometimes that's written as 1 minus p hat, just so you know the difference. So that would be 1 minus the 188 out of 523 to get a 64% of the students who have not participated in underage drinking. One thing just to always make sure your math is correct is that your p hat and your q hat should add up to 1, because they're complements of each other.

So I'm going to go ahead, I'm going to plug-in those values into my formula. So I have 0.36 for my p hat, 0.64 for my q hat, and my total sample was 523 students. Take the square root, and my standard error is 0.021.

In this example, we're still looking at the students who were surveyed about underage drinking, but notice how we've added on the proportion of underage drinkers nationally is 39%. We're still calculating the standard error of the sample proportion, but in this case, we know the population standard deviation, which is 39%. So we're going to use the formula pq over n square root. And we don't need to use p hat, which is the 188 out of 523 to make the estimate for our standard error. We actually know p, which is 39%, or 0.39.

So in this case, I'm going to use 0.39 for my p, and sometimes you'll see that written as p sub-0 and q sub-0 0. That's another way of indicating population proportion. And then the compliment to 39% would be 61%, because they have to add up to 1, or to 100.

And my sample size hasn't changed. It's still 523 students who were surveyed. We're going to take the square root, and our standard error is 0.021.

I hope this lesson was helpful in calculating standard error, and also helpful in practicing identifying which formula to use, based on the information you were given for these three formulas.