Estimation of Population Mean and Proportion in Statistics

 

Estimation of Population Mean and Proportion in Statistics

Introduction

In statistics, estimation is the process of inferring the value of a population parameter based on a sample drawn from the population. Two common parameters of interest are the population mean and the population proportion. Estimation can be done using point estimates or interval estimates. Point estimates provide a single value as an estimate, while interval estimates provide a range of values within which the parameter is expected to lie.

Estimation of Population Mean

Point Estimate of Population Mean

The sample mean ( $\bar{x}$ ) is the best point estimate of the population mean ( $\mu$ ).

Formula: $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

Example: Suppose we want to estimate the average height of students in a school. We randomly select a sample of 10 students and measure their heights (in cm): 150, 160, 165, 155, 170, 175, 160, 155, 165, 170.

Calculation: $\bar{x} = \frac{150 + 160 + 165 + 155 + 170 + 175 + 160 + 155 + 165 + 170}{10} = \frac{1625}{10} = 162.5$

The point estimate of the population mean height is 162.5 cm.

Interval Estimate of Population Mean

The interval estimate provides a range (confidence interval) within which the population mean is likely to lie. The confidence interval is calculated as:

$\bar{x} \pm Z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)$

where $\bar{x}$ is the sample mean, $Z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level, $\sigma$ is the population standard deviation, and $n$ is the sample size. If the population standard deviation ( $\sigma$ ) is unknown, the sample standard deviation ( $s$ ) is used and the $t$ -distribution is applied.

Example: Continuing with the previous example, assume the sample standard deviation $s = 8$ cm, and we want a 95% confidence interval. For a 95% confidence level, $Z_{\alpha/2} \approx 1.96$ .

Calculation: $\text{Margin of Error} = 1.96 \left(\frac{8}{\sqrt{10}}\right) = 1.96 \times 2.53 \approx 4.96$ $\text{Confidence Interval} = 162.5 \pm 4.96 = (157.54, 167.46)$

We are 95% confident that the population mean height lies between 157.54 cm and 167.46 cm.

Estimation of Population Proportion

Point Estimate of Population Proportion

The sample proportion ( $\hat{p}$ ) is the best point estimate of the population proportion ( $p$ ).

Formula: $\hat{p} = \frac{x}{n}$

where $x$ is the number of successes in the sample and $n$ is the sample size.

Example: Suppose we want to estimate the proportion of students in a school who prefer online classes. We survey a sample of 100 students, and 60 of them prefer online classes.

Calculation: $\hat{p} = \frac{60}{100} = 0.60$

The point estimate of the population proportion is 0.60, or 60%.

Interval Estimate of Population Proportion

The interval estimate for the population proportion is calculated as:

$\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

where $\hat{p}$ is the sample proportion, $Z_{\alpha/2}$ is the critical value from the standard normal distribution, and $n$ is the sample size.

Example: Continuing with the previous example, we want a 95% confidence interval. For a 95% confidence level, $Z_{\alpha/2} \approx 1.96$ .

Calculation: $\text{Margin of Error} = 1.96 \sqrt{\frac{0.60 \times 0.40}{100}} = 1.96 \sqrt{0.0024} \approx 1.96 \times 0.049 = 0.096$ $\text{Confidence Interval} = 0.60 \pm 0.096 = (0.504, 0.696)$

We are 95% confident that the population proportion of students who prefer online classes lies between 50.4% and 69.6%.

Conclusion

Estimating population parameters is a fundamental task in statistics. The sample mean and sample proportion provide the best point estimates for the population mean and population proportion, respectively. Interval estimates, such as confidence intervals, provide a range within which the population parameter is likely to lie, adding a measure of reliability to our estimates. Understanding and correctly applying these estimation techniques are essential for making informed decisions based on sample data.