# Confidence Intervals

## A Single Population Mean Using the Normal Distribution

A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of 
      and we have constructed the 90 percent confidence interval (5, 15), where the margin of error = 5. 

### Calculating the Confidence Interval

To construct a confidence interval for a single unknown population mean, μ, where the population standard deviation is known, we need 
  as an estimate for μ, and we need the margin of error. The margin of error for the population mean is called the error bound for a population mean (EBM). The sample mean, 
 is the point estimate of the unknown population mean, μ.
The confidence interval (CI) estimate will have the form:
(point estimate – error bound, point estimate + error bound) or, in symbols, (
 ).
The margin of error (EBM) depends on the confidence level (). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percentage of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, the person constructing the confidence interval will choose a confidence level of 90 percent or higher, because that person wants to be reasonably certain of his or her conclusions.
Another probability, which is called alpha 
 
   is related to the confidence level, CL. Alpha is the probability that the confidence interval does not contain the unknown population parameter. Mathematically, alpha can be computed as 
 
  .
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of 
  = 10, and we have constructed the 90 percent confidence interval (5, 15) where EBM = 5.
To get a 90 percent confidence interval, we must include the central 90 percent of the probability of the normal distribution. If we include the central 90 percent, we leave out a total of α = 10 percent in both tails, or 5 percent in each tail, of the normal distribution.
The critical value 1.645 is the  in a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. To capture the central 90 percent, we must go out 1.645 standard deviations on either side of the calculated sample mean. The critical value will change depending on the confidence level of the interval.
It is important that the standard deviation used be appropriate for the parameter we are estimating, so in this section, we need to use the standard deviation that applies to sample means, which is 
     . The fraction 
      is commonly called the standard error of the mean in order to distinguish clearly the standard deviation for a mean from the population standard deviation, σ.
2. 
      is normally distributed, that is, 
      ~ N
     .
3. When the population standard deviation 

### Calculating the Confidence Interval

To construct a confidence interval estimate for an unknown population mean, we need data from a random sample.  The steps to construct and interpret the confidence interval are as follows:
1.  Calculate the sample mean, 
      from the sample data. Remember, in this section, we already know the population standard deviation, σ.
2. Find the z-score that corresponds to the confidence level.
3. Calculate the error bound EBM.
4. Construct the confidence interval.
5. If we denote the critical z-score by 
, and the sample size by n, then the formula for the confidence interval with confidence level 
 
  , is given by 

6. Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)
We will first examine each step in more detail and then illustrate the process with some examples.

### Finding the z-Score for the Stated Confidence Level

When we know the population standard deviation, σ, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N(0, 1).
The confidence level, CL, is the area in the middle of the standard normal distribution. CL = 1 – α, so α is the area that is split equally between the two tails. Each of the tails contains an area equal to 
     .
The z-score that has an area to the right of 
      is denoted by 
     .
For example, when CL = 0.95, α = 0.05, and 
      = 0.025, we write 
      = 0.025.
The area to the right of z0.025 is 0.025 and the area to the left of z0.025 is 1 – 0.025 = 0.975.

 , using a calculator, computer, or standard normal probability table.
Normal table (see appendices) shows that the probability for 0 to 1.96 is 0.47500, and so the probability to the right tail of the critical value 1.96 is 0.5 – 0.475 = 0.025

### Calculating the Margin of Error EBM

The error bound formula for an unknown population mean, μ, when the population standard deviation, σ, is known is
Margin of error = 
 

### Constructing the Confidence Interval

The confidence interval estimate has the format
    sample mean plus or minus the margin of error.
The graph gives a picture of the entire situation
CL + 
      + 
      = CL + α = 1.

### Writing the Interpretation

The interpretation should clearly state the confidence level (CL), explain which population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints): "We estimate with ___percent confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units)."
Notice the difference in the confidence intervals calculated in  and the following Try It exercise. These intervals are different for several reasons: they are calculated from different samples, the samples are different sizes, and the intervals are calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.

### Changing the Confidence Level or Sample Size


### Working Backward to Find the Error Bound or Sample Mean

When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backward to find both the error bound and the sample mean.
2. From the upper value for the interval, subtract the sample mean,
3. Or, from the upper value for the interval, subtract the lower value.  Then divide the difference by 2.
2. Subtract the error bound from the upper value of the confidence interval,
3. Or, average the upper and lower endpoints of the confidence interval.
Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.

### Calculating the Sample Size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. In this situation, we are given the desired margin of error, EBM, and we need to compute the sample size n.
The formula for sample size is n = 
     , found by solving the error bound formula for n. Always round up the value of n to the closest integer.
In this formula, z is the critical value 
     , corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.

### References

Centers for Disease Control and Prevention. (n.d.). National health and nutrition examination survey. Retrieved from http://www.cdc.gov/nchs/nhanes.htm
Foothill De Anza Community College District. (n.d.). Headcount enrollment trends by student demographics ten-year fall trends to most recently completed fall. Retrieved from http://research.fhda.edu/factbook/FH_Demo_Trends/FoothillDemographicTrends.htm
Kuczmarski, R. J., et al. (2002). 2000 CDC growth charts for the United States: Methods and development. Retrieved from http://www.cdc.gov/growthcharts/2000growthchart-us.pdf
La, L., and; German, K. (n.d.). Cell phones with the highest radiation levels. CNET. Retrieved from http://reviews.cnet.com/cell-phone-radiation-levels/
U.S. Census Bureau. (n.d.). American FactFinder. Retrieved from http://factfinder2.census.gov/faces/nav/jsf/pages/searchresults.xhtml?refresh=t
U.S. Census Bureau. (2011). Mean income in the past 12 months (in 2011 inflation-adjusted dollars): 2011 American Community Survey 1-year estimates. Retrieved from http://factfinder2.census.gov/faces/tableservices/jsf/pages/productview.xhtml?pid=ACS_11_1YR_S1902&prodType=table
U.S. Federal Election Commission. (n.d.). Disclosure data catalog: Candidate summary report 2012. Retrieved from http://www.fec.gov/data/CandidateSummary.do?format=html&election_yr=2012
U.S. Federal Election Commission. (n.d.). Metadata description of candidate summary file. Retrieved from http://www.fec.gov/finance/disclosure/metadata/metadataforcandidatesummary.shtml

### Chapter Review

In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean (EBM). A confidence interval has the general form
(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM).
The calculation of EBM depends on the size of the sample and the level of confidence desired. The confidence level is the percentage of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding EBM increases as well. As the sample size increases, the EBM decreases. By the central limit theorem,

     
Given a confidence interval, you can work backward to find the error bound (EBM) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half of the difference of the upper and lower bounds.  To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.
Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the EBM formula for n to discover the size of the sample that is needed to achieve this goal:

### Formula Review


  The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.
The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by
    (lower bound, upper bound) = (point estimate – EBM, point estimate + EBM)

    EBM = 
      = the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.
CL = confidence level, or the proportion of confidence intervals created that is expected to contain the true population parameter
α = 1 – CL = the proportion of confidence intervals that will not contain the population parameter

      = the z-score with the property that the area to the right of the z-score is 
     ;	 this is the z-score, used in the calculation of EBM, where α = 1 – CL.
n = 
      = the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence
General form of a confidence interval
(lower value, upper value) = (point estimate error bound, point estimate + error bound)
To find the error bound when you know the confidence interval,
error bound = upper value point estimate or error bound = 
     
Single population mean, known standard deviation, normal distribution
Use the normal distribution for means; population standard deviation is known: EBM = z
     
The confidence interval has the format (
      − EBM, 
      + EBM).
Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 lb. We wish to construct a 95 percent confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 lb. The sample standard deviation is 11 lb.
Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.
Use the following information to answer the next 10 exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 lb, with a standard deviation of 0.1 lb. The population standard deviation is known to be 0.2 lb.
Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that 25 winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for winter Foothill College students. Let X = the age of a winter Foothill College student.
Construct a 95 percent confidence interval for the true mean age of winter Foothill College students  by working out and then answering the next eight exercises.

### Homework
