# 8.5: Chapter Formula Review

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• OpenStax
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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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