4.5.1 - Construct and Interpret the CI

4.5.1 - Construct and Interpret the CI

To construct a confidence interval we're going to use the following 3 steps:

  1. Step 1: Check Condition

    Check all conditions before using the sampling distribution of the sample proportion.

    We previously used \(np\) and \(n(1-p)\). But \(p\) is not known. Therefore, for the confidence interval, we will use:

    • \(n\hat{p}>5\) and
    • \(n(1-\hat{p})>5\)
  2. For a confidence interval for a proportion, there is a technique called exact methods. These methods can be used if the software offers it. These exact methods are more complicated and are based on the relationship between the binomial and another distribution we will later learn called the F-distribution. The Z-method is much simpler and fairly easy to compute. In fact, if you ever come across a published random survey (e.g. a Gallup poll) you can use the methods in this lesson to construct a reliable proportion confidence interval rather quickly.

    What can one do if the conditions are NOT satisfied?

  3. Step 2: Construct the General Form

    The general form of the confidence interval is '\(\text{point estimate }\pm M\times \hat{SE}(\text{estimate})\).' The point estimate is the sample proportion, \(\hat{p}\), and the estimated standard error is \(\hat{SE}(\hat{p})=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). If the conditions are satisfied, then the sampling distribution is approximately normal. Therefore, the multiplier comes from the normal distribution. This interval is also known as the one-sample z-interval for \(p\), or the Normal Approximation confidence interval for \(p\).

    \(\boldsymbol{\left(1-\alpha \right) 100\%}\) confidence interval for the population proportion, \(\boldsymbol{p}\)

    \(\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\)

    where \(z_{\alpha/2}\) represents a z-value with \(\alpha/2\) area to the right of it.

    General notes about the confidence interval...
    • The \(\pm\) in the formula above means "plus or minus". It is a shorthand way of writing
    • \((\hat{p}-z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p}+z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}})\)
    • It is centered at the point estimate, \(\hat{p}\).
    • The width of the interval is determined by the margin of error.
    • You must determine the multiplier.
  4. Step 3: Interpret the Confidence Interval

    Applying the template from earlier in the lesson we can say we are \((1-\alpha)100\%\) confident that the population proportion is between \(\hat{p}-z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) and \(\hat{p}+z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). The examples will go into more detail regarding the interpretation of the confidence interval.

 Think about it!
What terms in the margin of error would change the width of the confidence interval? Do the changes make it narrower or wider?


Construct a CI using Minitab

To construct a 1-proportion confidence interval...

  1. In Minitab choose Stat > Basic Statistics > 1 proportion .
  2. From the drop down box select the Summarized data option button. (If you have the raw data you would use the default drop down of One or more samples, each in a column.)
  3. Enter the number of successes in the Number of Events text box, and the sample size in the Number of Trials text box.
  4. Choose the Options button. The default confidence level is 95. If your desire another confidence level edit appropriately.
  5. To use the z- interval method choose Normal Approximation from the Method text box. The exact interval is always appropriate and is the default. Under the conditions that: $n \hat{p} \ge 5, n(1− \hat{p}) \ge 5$, one can also use the z-interval to approximate the answers. The exact interval and the z-interval should be very similar when the conditions are satisfied.
  6. Choose OK and OK again.

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