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7.3: Large Sample Estimation of a Population Proportion

  • Page ID
    565
  • [ "article:topic", "Confidence Intervals for a Proportion" ]

    Skills to Develop

    • To understand how to apply the formula for a confidence interval for a population proportion.

    Since from Section 6.3, we know the mean, standard deviation, and sampling distribution of the sample proportion  \(\hat{p}\) , the ideas of the previous two sections can be applied to produce a confidence interval for a population proportion. Here is the formula.

    Large Sample \(100(1−\alpha)\%\) Confidence Interval for a Population Proportion

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

    A sample is large if the interval  \([p-3\sigma_{\hat{p}},p+3\sigma _{\hat{p}}]\) lies wholly within the interval  \([0,1]\).

    In actual practice the value of \(p\) is not known, hence neither is  \(\sigma_{\hat{p}}\). In that case we substitute the known quantity  \(\hat{p}\)  for \(p\) in making the check; this means checking that the interval

    \[\left [ \hat{p}-3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\: \hat{p}+3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right ]\]

    lies wholly within the interval  \([0,1]\). 

    Example \(\PageIndex{1}\)

    To estimate the proportion of students at a large college who are female, a random sample of \(120\) students is selected. There are \(69\) female students in the sample. Construct a \(90\%\) confidence interval for the proportion of all students at the college who are female.

    Solution:

    The proportion of students in the sample who are female is 

    \[ \hat{p} =69/120=0.575\]

    Confidence level \(90\%\) means that  \(\alpha =1-0.90=0.10\) so  \(\alpha /2=0.05\). From the last line of Figure 7.1.6 we obtain \(z_{0.05}=1.645\).

    Thus

    \[\hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=0.575\pm 1.645\sqrt{\frac{(0.575)(0.425)}{120}}=0.575\pm 0.074\]

    One may be \(90\%\) confident that the true proportion of all students at the college who are female is contained in the interval  \((0.575-0.074,0.575+0.074)=(0.501,0.649)\).

    key takeaway

    • We have a single formula for a confidence interval for a population proportion, which is valid when the sample is large.
    • The condition that a sample be large is not that its size \(n\) be at least \(30\), but that the density function fit inside the interval \([0,1]\).

    Contributor

    • Anonymous