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Introduction to Confidence Intervals (page 248)
In chapter 7 we discussed how to make inferences about a population parameter based on a sample statistic. While this can be useful, it has severe limitations. In Chapter 8, we expand our toolbox to include Confidence Intervals. Instead of basing our inference on a single value, a point estimate, a Confidence Interval provides a range of values, an interval, which – at a certain level of confidence (90%, 95%, etc.) – contains the true population parameter. Having a range of values to make inferences about the population provides much more room for accuracy than making an inference off of only one value.
When we worked with probabilities based on sample means, we learned that there is only one population with many possible samples. With Confidence Intervals, we calculate a range of values based on one sample drawn in order to draw inferences about the population parameter (in this case, the population mean).
Confidence Intervals are made up of two parts, the point estimate and the margin of error, and they are constructed as: Point estimate ± Margin of error.
Thus, how big or small the Margin of Error is accounts for how large the range of possible values representing the population parameter will be. The Margin of error is made up of the desired confidence level and the standard error (which you learned about in chapter 7). The confidence level is represented as z/2 or t/2, where is the probability of error (you’ll see this in later chapters when we get into Hypothesis Testing) and is calculated as 100 – the confidence level. For example, if the confidence level is 95%, then is 5%. In the image below, the middle 95% of the curve represents the confidence interval – we are 95% confident that the true population parameter falls somewhere in that area. The two tails represent (/2).
Confidence Interval for the Population Mean when is known (page 248)
When we are working with quantitative
In chapter 7 we discussed how to make inferences about a population parameter based on a sample statistic. While this can be useful, it has severe limitations. In Chapter 8, we expand our toolbox to include Confidence Intervals. Instead of basing our inference on a single value, a point estimate, a Confidence Interval provides a range of values, an interval, which – at a certain level of confidence (90%, 95%, etc.) – contains the true population parameter. Having a range of values to make inferences about the population provides much more room for accuracy than making an inference off of only one value.
When we worked with probabilities based on sample means, we learned that there is only one population with many possible samples. With Confidence Intervals, we calculate a range of values based on one sample drawn in order to draw inferences about the population parameter (in this case, the population mean).
Confidence Intervals are made up of two parts, the point estimate and the margin of error, and they are constructed as: Point estimate ± Margin of error.
Thus, how big or small the Margin of Error is accounts for how large the range of possible values representing the population parameter will be. The Margin of error is made up of the desired confidence level and the standard error (which you learned about in chapter 7). The confidence level is represented as z/2 or t/2, where is the probability of error (you’ll see this in later chapters when we get into Hypothesis Testing) and is calculated as 100 – the confidence level. For example, if the confidence level is 95%, then is 5%. In the image below, the middle 95% of the curve represents the confidence interval – we are 95% confident that the true population parameter falls somewhere in that area. The two tails represent (/2).
Confidence Interval for the Population Mean when is known (page 248)
When we are working with quantitative