# High School: Statistics and Probability

### Making Inferences and Justifying Conclusions HSS-IC.B.4

4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Students should understand how the mean, proportion, and margin of error can all be gleaned from a sample survey. In order to do that, they have to know what these wonderful things are.

The population mean is the average value of the parameter for the whole population, and is expressed as μ. Students should know that it's nearly impossible to find the true value of μ, so we use the mean of our sample to estimate it. The larger of a sample we have, the closer we'll get to μ. We mean it.

Rather than finding μ, it may just be better to find the population's proportion. Population proportions can be given as point estimates (single values) or intervals (a range of values). That is, we can say that 80% of residents prefer candidate X, or that 75 - 85% prefer candidate X. Students should know when it's appropriate to use each one, depending on the accuracy of the study and the variability of the numbers.

Our sample estimate, randomly chosen, is nothing more than an estimate of our population mean or proportion. That means there will always be some margin of error involved. The only way we won't have a margin of error is if we had all the money and time in the world (yes, please!) to sample everyone in the population.

Before calculating this margin of error we must choose how confident we want to be about our results. For example, if we choose a 95% confidence interval (very standard) this means that out of 100 randomly chosen sample surveys, our result would be within our confidence interval 95 out of 100 times. Not too shabby.

For a 95% confidence interval our equation for the margin of error (where p is equal to the proportion in decimal form and N is our sample size) becomes:

But what if our results aren't in proportion form? How do we calculate margin of error for a mean score instead? That's when we use this equation (where N is our sample size and s is our standard deviation).

Armed with all this information, students should be able to emerge victorious in battles with population means, proportions, and margins of error.

#### Drills

1. A local candy store has found that kids prefer certain colors of candy regardless of their taste. For kids ages five to eight, the following data was collected:

 Blue 67 Red 89 Yellow 27 Green 13

What is the estimated population proportion of the most preferred candy color from this sample?

0.45

The most preferred color based on this sample survey is red. The population proportion is the frequency of red divided by the total sample size. Numerically, that's 89 divided by 196, which equals 0.45 (or 45%)

2. You've recently been volunteering your time helping the elderly play bingo at the local retirement home. (Strangely, you've noticed your cheeks are bruised, but that's probably due to all of the seniors pinching them.) You've randomly surveyed some of the bingo players to find that they play bingo a certain amount of hours per week. What is the estimated population mean based on your sample?

 Senior Citizen Hours Henry 4 Gertrude 3 Albert 5 Mary-Jo 12 Frank 14 Penelope 16

9 hours

The population mean is the average of the hours all the seniors spend playing bingo. Since we haven't sampled all the seniors, this random sample will have to suffice. That means we add up the hours and divide by the total number of players, or . That's the estimate of our population mean based on these six seniors.

3. Which of the following is a correct representation of a proportion?

A research firm that found that 85% of all employees spend an hour eating lunch

The answer for (A) is a population mean, not a proportion. For (B), a margin of error measures the certainty of the target parameter, but we don't know what that target parameter is. The only proportion given is in (C), since (D) is just an observational study without any numerical data.

4. A company specializing in building robots that clean your house has found that the average amount of time kids are forced (yes, forced) to spend cleaning their houses is about 2 hours per week. If their sample size was 1000 randomly chosen kids and the standard deviation was 0.3 hours, what is the margin of error for a confidence interval of 95%?

0.018

With a 95% confidence interval, we can use our formula  to find the margin of error. If we plug in s = 0.3 and N = 1000, we should get . That gives us a (B) as the right answer.

5. The ticket sales of the current top six movies, in figures of millions, are shown below. Assuming a 95% confidence interval, what is the margin of error for the average number of ticket sales?

 Movie Ticket Sales (in millions) Are You Afraid of the Dark Knight? 6.7 Apollo 13 Going on 30 11.3 The Sixth Sense and Sensibility 38.5 The Godfather of the Bride 100.4 Once Upon a Forrest Gump 76.3 The Breakfast Fight Club 16

4.79

Before we can find any margin of error, we need to calculate the mean and standard deviation of the ticket sales. The total number of ticket sales (N) is 249.2, our average is 41.5, and our standard deviation is 38.63. Using the formula , we can see that our margin of error is 4.79. Of course, these numbers are all in millions.

6. Of all the new 3D television users, 72% have reported not having any side effects such as vertigo, headaches, or undue eyestrain as compared to watching regular television. If this sample questionnaire had 2500 respondents, what is the margin of error? Round to the nearest hundredth decimal.

0.02

First figure out what the question is asking, and which equation must be used. We are given the number of respondents (2500) and the proportion (0.72). We can plug the numbers into the equation:

If we do, we end up with:

which equals 0.017, or about 0.02.

7. The top 100 male weightlifters were tested on the bench press. They averaged 515 pounds for one repetition. What is the margin of error if the standard deviation was 52 pounds (assuming a 95% confidence interval)?

10 lbs

We can plug our values of N = 100 and s = 52 into the equation  to get a value of 10.2, or about 10. That means our answer is (C).

8. A total of 50 boats participate in a race. You decide to figure out what the average finishing time is, but everything happens so fast, and you can only get a sample of five boats' times written down. The average finishing times (in seconds) are given below. What is the population average based on your sample?

 Boat Seconds 1 63 2 62 3 67 4 69 5 73

66.8

The average of our sample will have to be good enough to estimate population average. In order to find the average, all we need to do is divide the sum of all the finishing times by 5. That will give us 66.8 seconds, so (B) is right.

9. Which of the following is most like a proportion of a sample?

All of the above

All of those are proportions, just written out a bit differently. A proportion is usually given as a comparison of two values. (Even a percent is a comparison of the percentage to a total of 100.) Nothing too hard about that, is there?

10. Which of the following best represents a sample mean?