Estimation

Japanese school year: Math B

What you learn

You will learn how to estimate population means and proportions within intervals based on a specified confidence level using sample statistics. This technique is indispensable in opinion polling and industrial quality control. A solid understanding of the normal distribution and sample means will be beneficial.

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Key points

This formula shows how the variance of the sample mean $\bar{X}$ shrinks as sample size $n$ increases. Dividing the population variance $\sigma^2$ by $n$ indicates that larger samples produce a more stable average.

V(\bar{X}) = \frac{\sigma^2}{n}

Use this formula to estimate the range that contains the true population mean $m$ with 95% confidence (95% confidence interval). It centers on the sample mean $\bar{X}$ with an error margin based on sample size $n$ and standard deviation $\sigma$ .

\bar{X} - 1.96 \frac{\sigma}{\sqrt{n}} \le m \le \bar{X} + 1.96 \frac{\sigma}{\sqrt{n}}

Use this formula to estimate the true population proportion $p$ with 95% confidence. It calculates a confidence interval around the observed sample proportion $R$ using sample size $n$ .

R - 1.96 \sqrt{\frac{R(1-R)}{n}} \le p \le R + 1.96 \sqrt{\frac{R(1-R)}{n}}

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Practice 1Coming soon