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8.8: Formula Review

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    8.3: A Confidence Interval When the Population Standard Deviation Is Unknown and Small Sample Case

    \(s=\) the standard deviation of sample values.

    \(t=\dfrac{\bar{x}-\mu}{\dfrac{s}{\sqrt{n}}}\) is the formula for the \(t\)-score which measures how far away a measure is from the population mean in the

    Student's t-distribution

    \(d f=n-1\); the degrees of freedom for a Student's t -distribution where n represents the size of the sample

    \(T \sim t_{d f}\) the random variable, \(T\), has a Student's t -distribution with \(d f\) degrees of freedom

    The general form for a confidence interval for a single mean, population standard deviation unknown, and sample size less than 30 Student's t is given by: \(\bar{x}-t_{\mathrm{v}, \alpha}\left(\dfrac{s}{\sqrt{n}}\right) \leq \mu \leq \bar{x}+t_{\mathrm{v}, \alpha}\left(\dfrac{s}{\sqrt{n}}\right)\)

    8.4: A Confidence Interval for A Population Proportion

    \(p^{\prime}=\dfrac{x}{n}\) where \(x\) represents the number of successes in a sample and \(n\) represents the sample size. The variable \(p^{\prime}\) is the sample proportion and serves as the point estimate for the true population proportion.

    \[q^{\prime}=1-p^{\prime}\]

    The variable \(p^{\prime}\) has a binomial distribution that can be approximated with the normal distribution shown here. The confidence interval for the true population proportion is given by the formula:

    \[\mathrm{p}^{\prime}-Z_\alpha \sqrt{\dfrac{\mathrm{p}^{\prime} \mathrm{q}^{\prime}}{n}} \leq p \leq \mathrm{p}^{\prime}+Z_\alpha \sqrt{\dfrac{\mathrm{p}^{\prime} \mathrm{q}^{\prime}}{n}}\]

    \(n=\dfrac{Z_{\dfrac{\alpha}{2}}{ }^2 p^{\prime} q^{\prime}}{e^2}\) provides the number of observations needed to sample to estimate the population proportion, \(p\), with confidence \(1-\alpha\) and margin of error \(e\). Where \(e=\) the acceptable difference between the actual population proportion and the sample proportion.

    8.5: Calculating the Sample Size n: Continuous and Binary Random Variables

    \(n=\dfrac{Z^2 \sigma^2}{(\bar{x}-\mu)^2}=\) the formula used to determine the sample size \((n)\) needed to achieve a desired margin of error at a given level of confidence for a continuous random variable

    \(n=\dfrac{Z_\alpha^2 \mathrm{pq}}{e^2}=\) the formula used to determine the sample size if the random variable is binary

    8.8: Formula Review is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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