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7.1: Introduction to Confidence Intervals

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    Learning Objectives
    • Estimate population parameters (e.g., mean or proportion) using confidence intervals based on sample statistics.
    • Provide a range that likely contains the true population value at a specified confidence level (e.g., 95%).
    • Measure the reliability of estimates and account for sampling variability in concluding data.

    Point Estimate:

    A point estimator is a statistic that is calculated from a sample. As an example, when you want to estimate the population mean, \(\mu\), the point estimator is the sample mean, \(\overline{x}\). To estimate the population proportion, p, you use the sample proportion, \(\hat{p}\).

    Point estimators are easy to calculate, but they have some drawbacks. First, if the sample size is large, then the estimate is better. However, the sample size is not always given for a point estimator. Also, it is challenging to measure the accuracy of the forecast. Both of these problems are solved with a confidence interval.

    Definition \(\PageIndex{1}\)

    Confidence interval: This is where you have an interval surrounding your parameter, and the interval has a chance of being a true statement. In general, a confidence interval looks like this: The standard formula is Point Estimate \( \pm E\), where E is the margin of error term that is added and subtracted from the point estimator. Thus making an interval.

    Interpreting a confidence interval:

    The statistical interpretation is that the confidence interval has a probability (1 - \(\alpha\), where \(\alpha\) is the complement of the confidence level) of containing the population parameter. For example, if there is a 95% confidence interval of 0.65 < p < 0.73, then it can be said that “there is a 95% chance that the interval 0.65 to 0.73 contains the true population proportion.” This means that for 100 intervals, 95 of them will contain the true proportion, and 5% will not. The wrong interpretation is that there is a 95% chance that the true value of p will fall between 0.65 and 0.73. This interpretation is inaccurate because the true value (called the parameter) is fixed. Fixed values are not random and do not have probabilities associated with it. Only the interval is random. In the end, the intervals are randomly computed to capture the parameter with the interval. So the probability is the chance that each interval captures a parameter, not that the parameter falls in the interval.

    The common probabilities utilized to compute the confidence intervals are 90%, 95%, and 99%. These are known as the confidence levels. The confidence level and the alpha level are related. For a two-tailed test, the confidence level is C = 1 - \(\alpha\). This is because the \(\alpha\) is both tails and the confidence level is the area between the two tails. As an example, for a two-tailed test ( \(H_1\) is not equal to) with \(\alpha\) equal to 0.10, the confidence level would be 0.90 or 90%. If you have a one-tailed test, then your \(\alpha\) is only one tail. Because of symmetry, the other tail is also \(\alpha\). So you have 2\(\alpha\) with both tails. So the confidence level, which is the area between the two tails, is C = 1 - 2\(\alpha\). A visual of the probability distribution using a normal distribution is provided below.

    Graph of distribution with areas for confidence level and alpha.

    Figure \(\PageIndex{1}\): Confidence Level and \(\alpha\) Over 2 Areas.
    Example \(\PageIndex{1}\) stating the statistical and real world interpretations for a confidence interval
    1. Suppose the confidence level is 95%. The confidence interval for the mean age a woman gets married in 2013 is \(26<\mu<28\). State the statistical and real-world interpretations of this statement.
    2. Suppose a 99% confidence interval for the proportion of Americans who have tried marijuana as of 2013 is \(0.35<p<0.41\). State the statistical and real-world interpretations of this statement
    Solution
    1. Statistical Interpretation: There is a 95% chance that the interval \(26<\mu<28\) contains the mean age a woman gets married in 2013.
      Real World Interpretation: The mean age that a woman married in 2013 is between 26 and 28 years of age.
    2. Statistical Interpretation: There is a 99% chance that the interval \(0.35<p<0.41\) contains the proportion of Americans who have tried marijuana as of 2013. Real World Interpretation: The proportion of Americans who have tried marijuana as of 2013 is between 0.35 and 0.41.

    One final thing to understand about confidence is how the sample size and confidence level affect how wide the interval is. The following discussion demonstrates what happens to the width of the interval as you get more confident.

    Think about shooting an arrow into the target. Suppose a person is good at shooting arrows and has a 90% chance of hitting the bull’s eye. Keep in mind that the bull’s eye is small. Since the person hits the bull’s eye approximately 90% of the time, the person will probably hit inside the next ring out 95% of the time. The person will have a better chance if the bull's eye can be increased. In such a situation, the person may have a 99% chance of hitting the target, but that is a much bigger circle to hit. As your confidence in hitting the target increases, the circle you hit gets bigger. The same is true for confidence intervals. This is demonstrated in Figure \(\PageIndex{1}\).

    Affect of Confidence Level on Width
    Figure \(\PageIndex{2}\): Affect of Confidence Level on Width

    The higher level of confidence makes a wider interval. There’s a trade-off between width and confidence level. You can be confident about your answer but the answer will not be precise. Or you can have a precise answer (small margin of error) but not be very confident about your answer.

    Now look at how the sample size affects the size of the interval. Suppose Figure \(\PageIndex{2}\) represents confidence intervals calculated on a 95% interval. A larger sample size from a representative sample makes the width of the interval narrower. This makes sense. Large samples are closer to the true population so the point estimate is close to the true value.

    Affect of sample size on width.
    Figure \(\PageIndex{3}\): Affect of Sample Size on Width

    Now that the foundations underlying confidence intervals have been introduced. The next step is to demonstrate how to calculate them using a formula. The formula depends on which parameter is being estimated, the given confidence level, and other information.

    Exercises

    1. Suppose you compute a confidence interval with a sample size of 25. What will happen to the confidence interval if the sample size increases to 50?

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    1. Suppose you compute a 95% confidence interval. What will happen to the confidence interval if you increase the confidence level to 99%?

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    1. A 95% confidence interval is 6353 km < \( \mu \) < 6384 km, where is the mean diameter of the Earth. State the statistical interpretation.

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    Answers

    If you are an instructor and want the solutions to all the exercise questions for each section, please email Toros Berberyan.

    This page titled 7.1: Introduction to Confidence Intervals is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Toros Berberyan, Tracy Nguyen, and Alfie Swan via source content that was edited to the style and standards of the LibreTexts platform.