Statistics Confidence Intervals for the Mean – Large Samples
There are two types of estimation.
Let’s say that you take your car to your local car dealer’s service department and you ask how much it will cost to fix your car. If they say it will cost you $500 then they are providing a point estimate. If they say it will cost somewhere between $400 and $600 then they are providing an interval estimate.
Point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean Îź is the sample mean x . The sample mean is not the same thing as the population mean. It could be an underestimate or an overestimate.
As long as the sample size is large enough, you can use the sample mean as the population mean.
Example: Here are the IQ test scores of 31 seventh-grade girls in a school district:
114, 100, 104, 89, 102, 91, 114, 114, 103, 105, 108, 130, 120, 132, 111, 128, 118, 119, 86, 72, 111, 103, 74, 112, 107, 103, 98, 96, 112, 112, 93 Treat the 31 girls as a simple random sample of all seventh-grade girls in the school district.
Find a point estimate of the population mean Âľ.
x  105.8387 So, the point estimate for the mean IQ test score of all seventh-grade girls in this schools district is 105.8387.
Interval estimate is an interval or range of values used to estimate a population parameter. Interval estimates are desirable because the estimate of the mean varies from sample to sample. The interval estimate gives an indication of how much uncertainty there is in the estimate of the true mean.
Level of confidence (c) is the probability that the interval estimate contains the population parameter. In this course, 90%, 95%, and 99% levels of confidence will be used most often.
The following z-scores correspond to certain levels of confidence. You can get these in your textbook 90% level of confidence zc = 1.645 95% level of confidence zc = 1.96 99% level of confidence zc = 2.576 I like to be more precise. Here’s how. 1 1 90% level of confidence 1 .9 .1 .05 2 2
Maximum error of estimate (E), sometimes called the margin of error or error tolerance, is the greatest possible distance between the point estimate and the value of the parameter it is estimating. E zc x zc
n
When n≥30, s may be used instead of σ.
Example: Use the IQ scores of our seven grade girls. Treat the 31 girls as a simple random sample of all seventh-grade girls in the school district. Use a 95% confidence level to find the maximum error of estimate for the mean IQ test score of the population mean µ. You don’t know the population standard deviation σ. But since n ≥ 30, you can use s in place of σ. x 105.8387 s = 14.2714
Maximum error of estimate (E) E zc
14.2714 1.9600 5.0239 n 31
Using a point estimate and a maximum error of estimate, we can construct an interval estimate of a population parameter. This interval estimate is called a confidence interval. Confidence interval: x E x E
The probability that the confidence interval contains µ is c.
Guidelines: Finding a confidence interval for a population mean (n ≥ 30 or σ known with a normally distributed population) 1. Find the sample mean 2. If you have σ , great! Otherwise, if n≥30 , find the sample standard deviation s and use it as an estimate for σ . 3. Determine the critical value zc that corresponds to the desired level of confidence. 4. Find the maximum error of estimate E. 5. Find the left and right endpoints and form the confidence interval.
The confidence level describes the uncertainty associated with a sampling method. Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter. A 95% confidence level means that 95% of the intervals would include the parameter.
Example: Use the IQ scores of our seven grade girls. Treat the 31 girls as a simple random sample of all seventh-grade girls in the school district. Use a 95% confidence level to find the maximum error of estimate for the mean IQ test score of the population mean µ. You don’t know the population standard deviation σ. But since n ≥ 30, you can use s in place of σ. x 105.8387 s = 14.2714
95% Confidence Interval 14.2714 E zc 1.9600 5.0239 n 31
x E 100.8148, 110.8626
So with 95% confidence, we can say that the interval from 100.8148 and 110.8626 contains the population mean IQ score for all seventh-grade girls in this school district.
So with 95% confidence, we can say that the interval from 100.81 and 110.86 contains the population mean IQ score for all seventh-grade girls in this school district. When we form a confidence interval for Âľ, we express our confidence in the interval.
So why not just use 99% confidence intervals rather than 95% intervals? The answer is that for a given sample size, the 99% confidence interval will be wider, therefore less precise than a 95% confidence interval. Why do you think a 99% confidence interval is wider than a 70% confidence interval or a 50% confidence interval? As the level of confidence increases, if the sample stays the same, the confidence interval widens. When this happens, the precision of the estimate decreases. One way to improve the precision of an estimate without decreasing the level of confidence is to increase the sample size.
70% confidence interval (103.18, 108.5)
80% confidence interval (102.55, 109.12)
90% confidence interval (101.62, 110.05)
99% confidence interval (99.236, 112.44)
Notice as the level of confidence increases, the interval widens.
The size of the sample needed for a desired level of confidence zc n E
2
Example: How large of a sample of seventh-grade girls from the previous examples would be needed to estimate the mean IQ score within ± 5 points with 99% confidence. s = 14.2714 E=5 z.99 = 2.5758
2
zc 2.575 14.2714 n 54.0527 5 E 2
Round up to obtain a whole number. So, we should include at least 55 seventh-grade girls from the district in our sample.