• Using the mtcars data we will give the best guess of average gas mileage.

• Goal: give the best guess of what the average gas mileage is and check the deviation.

• Point estimate: The sample mean
• Confidence interval: a range around that guess.

Sample mean:

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\]

Sample standard deviation:

\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2 \qquad = \sqrt{s^2}\]

\[\bar{x} \;\pm\; t^* \cdot \frac{s}{\sqrt{n}}\]

• \(\dfrac{s}{\sqrt{n}}\) is the standard error, the typical error of the sample mean.
• \(t^*\) is a multiplier from the t-distribution, about 2 for a 95% interval.
• The part after the \(\pm\) is the margin of error.

Bigger \(n\) makes the margin of error smaller. Bigger \(s\) makes it larger.

x    <- mtcars$mpg
n    <- length(x)

xbar <- mean(x)                  
s    <- sd(x)                    
se   <- s / sqrt(n)              
tcrit <- qt(0.975, df = n - 1)   

c(xbar - tcrit * se, xbar + tcrit * se)   

t.test(x)$conf.int               

Conclusion: most cars fall between 15 and 25 mpg, and the mean sits near the middle.

Conclusion: the 4-cylinder and 8-cylinder intervals do not overlap, so their average mileage is different.

Displays margin of error, standard deviation and the sample size

• Sample mean: 20.09 mpg
• Sample standard deviation: 6.03 mpg
• Standard error: 1.07 mpg
• Margin of error: 2.17 mpg
• 95% confidence interval: (17.92, 22.26) mpg