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8.5: Chapter Formula Review

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    A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case

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

    \(t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) 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

    \(df = n - 1\); 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 df 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: \(\overline{x}-t_{\mathrm{v}, \alpha}\left(\frac{s}{\sqrt{n}}\right) \leq \mu \leq \overline{x}+t_{\mathrm{v}, \alpha}\left(\frac{s}{\sqrt{n}}\right)\)

    A Confidence Interval for A Population Proportion

    \(p^{\prime}=\frac{x}{n}\) where \(x\) represents the number of successes in a sample and \(n\) represents the sample size. The variable p′ 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:

    \(\mathrm{p}^{\prime}-Z_{\alpha} \sqrt{\frac{\mathrm{p}^{\prime} \mathrm{q}^{\prime}}{n}} \leq p \leq \mathrm{p}^{\prime}+Z_{\alpha} \sqrt{\frac{\mathrm{p}^{\prime} \mathrm{q}^{\prime}}{n}}\)

    \(n=\frac{Z_{\frac{\alpha}{2}}^{2} p^{\prime} q^{\prime}}{e^{2}}\) 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.

    Calculating the Sample Size n: Continuous and Binary Random Variables

    \(n=\frac{Z^{2} \sigma^{2}}{(\overline{x}-\mu)^{2}}\) = 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

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

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