7.13 Margin of Error
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- Gilbert Malone
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1 7.13 Margin of Error Objectives: 1. Recognize the meaning of margin of error (given a margin of error) in th estimates. 2. Explain that larger sample sizes lead to a smaller margin of error. 3. All other aspects being unchanged, using a larger sample size will decrease the margin of error. Recall the Random Rectangles Activity. We took random samples of 5 and 10 rectangles and saw that these random samples tended to estimate the average area of the rectangles better than when we guessed or just used our judgment. Big Conclusion #1: Random samples let us make valid estimates of the population mean. 1
2 μ μ is a fixed value, the mean of the population. One Big Conclusion #2 (part a): Different samples give us different sample means. x for each sample Notice that x can vary. If were asking a numerical (quantitative) question, we find the sample mean. One for each sample Notice that can vary. If were asking a yes/no (categorical) question, we find the sample p proportion. p is a fixed value, the proportion of the population with some quality. Big Conclusion #2 (part b): Different samples give us different sample proportions. 2
3 These dotplots show the averages that one teacher's classes reported for samples of 5, 10, 20, and 50 rectangles. The actual average area of the 100 rectangles is μ=7.42 (marked with the triangles). As the sample size increases, what happens to the variation in their estimates (averages)? Sample size affects how much our statistics vary. Smaller samples Larger samples Statistics vary quite a bit from Statistics tend to be close to the population parameter and the population parameter from one another. and to one another. Big Conclusion #3: Larger samples lead to statistics that are closer to the population parameter we're estimating. 3
4 Example 1: Each student in a class took a random sample of 10 kids from their school and calculated the average amount of time spent on Instagram daily. Each student in another class did the same, but with samples of size 30. Which class will have sample means that are closer to one another? One statistic from each sample parameter Even though the statistics vary from the parameter and from one another, there's a limit on how far above or below the parameter that we expect the statistic to be. That maximum distance we expect a statistic to vary from the parameter is called the margin of error. 4
5 A margin of error is loosely defined as the largest expected size of the difference between an estimate (sample statistic) and the actual population value (parameter) that is being estimated. "margin of error" is also called the "maximum error of the estimate" The margin of error is the largest expected distance between any statistic and the actual population parameter that is being estimated. The graph shows the averages from 100 different random samples. Statistics tells us that we could estimate the margin of error by taking the distance between the largest and smallest sample means, and then dividing by 2. So, = 4.1 = The average for a sample can be expected to differ from the average of the population by about 2.05 and this 2.05 is the margin of error. 5
6 Big Conclusion #4: The margin of error is how far we expect a population parameter to be from our sample statistic. Example 2: A biologist measured the length of a random sample of fish from a lake and found this: the sample mean was 9.2 inches the margin of error was 1.3 inches Explain what the margin of error 1.3 inches means. 6
7 Example 3: A biologist examined a random sample of fish from a lake and found this: the sample proportion was 34% bluegill fish the margin of error was 3.5%. Explain what the margin of error 3.5% means. We don't want to take lots and lots of samples in order to calculate the margin of error. By the time we do that, we might as well have just taken a census of the whole population. Luckily, we have these formulas for calculating the margin of error from just one sample. For categorical data (yes/no questions): For quantitative data (numerical questions): 7
8 For categorical data (yes/no questions): For quantitative data (numerical questions): In both formulas, the sample size (n) is in the denominator, so a larger sample size gives us a smaller margin of error. n = sample size p hat = sample proportion s = sample standard deviation The "2" in each formula comes from The Rule. Big Conclusion #5: We have formulas for the margin of error that are based on the rule and a larger sample leads to a smaller margin of error. Example 4: A random sample in a certain high school asked 200 smartphone owning students whether they owned an iphone. It turned out that 76% of smartphone owning students in the sample had iphones. Choose the correct formula and then calculate the margin of error. 8
9 Example 5: In 2002, a random sample of 10 midsize luxury cars was tested in a 5 mph front end crash into a flat barrier. The average cost of the repairs for the 10 cars was $ and the standard deviation was $ Choose the correct formula and then calculate the margin of error. 9
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