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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:
We will first examine each step in more detail, and then illustrate the process with some examples.
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 $\frac{\alpha}{2}$ .
The z-score that has an area to the right of $\frac{\alpha}{2}$ is denoted by ${Z}_{\frac{\alpha}{2}}$ .
For example, when CL = 0.95, α = 0.05 and $\frac{\alpha}{2}$ = 0.025; we write ${Z}_{\frac{\alpha}{2}}$ = ${Z}_{0.025}$ .
The area to the right of ${Z}_{0.025}$ is 0.025 and the area to the left of ${Z}_{0.025}$ is 1 – 0.025 = 0.975.
${Z}_{\frac{\alpha}{2}}\text{=}{Z}_{0.025}\text{=1}\text{.96}$ , using a standard normal probability table. We will see later that we can use a different probability table, the Student's t-distribution, for finding the number of standard deviations of commonly used levels of confidence.
The error bound formula for an unknown population mean μ when the population standard deviation σ is known is
The graph gives a picture of the entire situation.
CL + $\frac{\alpha}{2}$ + $\frac{\alpha}{2}$ = CL + α = 1.
The interpretation should clearly state the confidence level ( CL ), explain what population parameter is being estimated (here, a population mean ), and state the confidence interval (both endpoints). "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units)."
Suppose we are interested in the mean scores on an exam. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68 ( $\stackrel{-}{X}$ = 68). In this example we have the unusual knowledge that the population standard deviation is 3 points. Do not count on knowing the population parameters outside of textbook examples. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).
Find a 90% confidence interval for the true (population) mean of statistics exam scores.
To find the confidence interval, you need the sample mean, $\stackrel{-}{x}$ , and the EBM .
CL = 0.90 so α = 1 – CL = 1 – 0.90 = 0.10
$\frac{\alpha}{2}$ = 0.05 ${Z}_{\frac{\alpha}{2}}={z}_{0.05}$
The area to the right of Z _{0.05} is 0.05 and the area to the left of Z _{0.05} is 1 – 0.05 = 0.95.
${Z}_{\frac{\alpha}{2}}\text{=}{Z}_{0.05}\text{=1}\text{.645}$
This can be found using a computer, or using a probability table for the standard normal distribution. Because the common levels of confidence in the social sciences are 90%, 95% and 99% it will not be long until you become familiar with the numbers , 1.645, 1.96, and 2.56
EBM = (1.645) $\left(\frac{3}{\sqrt{36}}\right)$ = 0.8225
$\stackrel{-}{x}$ - EBM = 68 - 0.8225 = 67.1775
$\stackrel{-}{x}$ + EBM = 68 + 0.8225 = 68.8225
The 90% confidence interval is
(67.1775, 68.8225).
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