Point estimation is the process of using sample data to estimate an unknown population parameter. Common point estimates include:

• Sample Mean (\(\hat{\mu}\)): A point estimate for the population mean.

• Sample Proportion (\(\hat{p}\)): A point estimate for the population proportion.

The mean of MPG

As an example, we will use the mtcars dataset to estimate the population mean of the miles per gallon (mpg) for cars.

Sample Mean Estimation

Based on the 32 cars in the dataset, the point of estimate for the population mean of miles per gallon is 20.09.

sample_mean_mpg <- mean(mtcars$mpg)
sample_mean_mpg

## [1] 20.09062

A point estimator is unbiased if the expected value of the estimator equals the true value of the parameter.

For the sample mean of mtcars:

The sample mean is an unbiased estimator of the population mean. Thus, we expect that the sample mean we calculated, 20.09, is close to the true population mean.

Relationship between miles per gallon (mpg), weight (wt), and horsepower (hp) of the cars.

The 95% confidence interval for the population mean is calculated using the formula: \[ \hat{\mu} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \]

Calculating 95% confidence interval for mpg

t.test(mtcars$mpg)$conf.int

## [1] 17.91768 22.26357
## attr(,"conf.level")
## [1] 0.95

The standard error (SE) of the mean gives us an idea of how precise our sample mean is. It is the standard deviation of the sampling distribution of the sample mean. We calculate it as:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

where:

• \(\sigma\) is the standard deviation of the sample, and
• \(n\) is the sample size.