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7.5: Formulas for Chapter 7

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    Formulas for Chapter 7

    Confidence Interval for the Mean When \(\sigma\) is Known

    A confidence interval of the mean using z-values is a range of values, based on sample data, that is likely to contain the true population mean with a certain level of confidence, such as 95%. This approach is used when the population standard deviation is known and the sample size is sufficiently large (typically n ≥ 30). The confidence interval is constructed using the sample mean, the z-score corresponding to the desired confidence level, and the standard error of the mean.

    \(\bar{x} - z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}} < \mu < \bar{x} + z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}}\)

    Where:

    • \( \bar{x} \) is the sample mean
    • \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level
    • \( \sigma \) is the population standard deviation
    • \( n \) is the sample size
    • \( \mu \) is the population mean
    • \( \frac{\sigma}{\sqrt{n}} \) is the standard error of the mean

    Confidence Interval for the Mean When \(\sigma\) is Not Known

    A confidence interval for the mean using t-values is used when the population standard deviation is unknown and the sample size is relatively small, typically less than 30. In this case, the sample standard deviation is used instead of the population standard deviation, and the t-distribution is used instead of the normal distribution. This method provides a range of values that is likely to contain the true population mean with a specified level of confidence.

    \(\bar{x} - t_{\alpha/2} \cdot \dfrac{s}{\sqrt{n}} < \mu < \bar{x} + t_{\alpha/2} \cdot \dfrac{s}{\sqrt{n}}\)

    Where:

    • \( \bar{x} \) is the sample mean
    • \( t_{\alpha/2} \) is the t-score for the desired confidence level and degrees of freedom
    • \( s \) is the sample standard deviation
    • \( n \) is the sample size
    • \( \mu \) is the population mean
    • \( \frac{s}{\sqrt{n}} \) is the standard error of the mean

    Sample Proportion and Complement

    The sample proportion of success, denoted as \(\hat{p}\), is the ratio of the number of successes in the sample to the total number of observations. It is calculated using the formula \(\hat{p} = \dfrac{X}{n}\), where \(X\) is the number of observed successes and \(n\) is the sample size. The complement of the sample proportion, denoted as \(\hat{q}\), represents the proportion of failures and is given by \(\hat{q} = 1 - \hat{p}\).

    Confidence Interval for Proportion

    A confidence interval for proportions is used to estimate the true population proportion based on a sample proportion. It provides a range of values within which the population proportion is likely to fall, with a certain level of confidence. This method is appropriate when the sample size is large enough such that both \(n\hat{p}\)​ and \(n\hat{q}\) are at least 5, where \(\hat{q} = 1 - \hat{p}\). The normal approximation to the binomial distribution is used to construct the interval.

    \(\hat{p} - z_{\alpha/2} \cdot \sqrt{\dfrac{\hat{p}\hat{q}}{n}} < p < \hat{p} + z_{\alpha/2} \cdot \sqrt{\dfrac{\hat{p}\hat{q}}{n}}\)

    Where:

    • \( \hat{p} \)is the sample proportion
    • \( \hat{q} = 1 - \hat{p} \) is the complement of the sample proportion
    • \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level
    • \( n \) is the sample size
    • \( p \) is the population proportion
    • \( \sqrt{\dfrac{\hat{p}\hat{q}}{n}} \) is the standard error of the proportion

    Minimum Sample Size for a Confidence Interval for the Mean

    The formula for minimum sample size is used to determine how many observations are needed to estimate a population mean within a specified margin of error at a certain confidence level. This ensures that the resulting confidence interval is both accurate and reliable. The formula takes into account the desired precision, the variability in the population, and the confidence level.

    \(n = \left( \dfrac{z_{\alpha/2} \cdot \sigma}{E} \right)^2\)

    Where:

    • \( n \) is the minimum required sample size
    • \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level
    • \( \sigma \) is the population standard deviation
    • \( E \) is the desired margin of error

    Minimum Sample Size for a Confidence Interval for a Proportion

    The formula for the minimum sample size needed to estimate a population proportion is used to calculate how many observations are required to achieve a desired margin of error at a specific confidence level. It helps ensure the estimate of the population proportion is precise and reliable.

    \(n = \hat{p} \cdot \hat{q} \cdot \left( \dfrac{z_{\alpha/2}}{E} \right)^2\)

    Where:

    • \( n \) is the minimum required sample size
    • \( \hat{p} \) is the sample proportion (estimated proportion of success)
    • \( \hat{q} = 1 - \hat{p} \) is the complement of the sample proportion (estimated proportion of failure)
    • \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level
    • \( E \) is the desired margin of error

    This page titled 7.5: Formulas for Chapter 7 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Toros Berberyan, Tracy Nguyen, and Alfie Swan.