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8.8: Formula Review

Last updated

Jun 26, 2024
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- 8.7: Chapter Review
- 8.9: Practice

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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

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

A Confidence Interval for A Population Proportion

Calculating the Sample Size n: Continuous and Binary Random Variables

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