problem 4: this is the data from a sample: 8, 2, 5, 4, 11 and 9. find the sample average (the sample mean)…

Explanation:

Step1: Identify the sample data and size

The sample data is ${8, 2, 5, 4, 11, 9}$. The sample size $n$ is $6$.

Step2: Calculate the sample mean

Sum the data points and divide by the sample size. $$\bar{x} = \frac{\sum x_i}{n} = \frac{8 + 2 + 5 + 4 + 11 + 9}{6} = \frac{39}{6} = 6.5$$

Step3: Calculate the sum of squared deviations

Subtract the mean from each value, square it, and sum. $$\sum (x_i - \bar{x})^2 = (8-6.5)^2 + (2-6.5)^2 + (5-6.5)^2 + (4-6.5)^2 + (11-6.5)^2 + (9-6.5)^2$$ $$= (1.5)^2 + (-4.5)^2 + (-1.5)^2 + (-2.5)^2 + (4.5)^2 + (2.5)^2$$ $$= 2.25 + 20.25 + 2.25 + 6.25 + 20.25 + 6.25 = 57.5$$

Step4: Calculate the sample variance

Divide the sum of squared deviations by $n - 1$. $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{57.5}{6 - 1} = \frac{57.5}{5} = 11.5$$

Step5: Calculate the sample standard deviation

Take the square root of the sample variance. $$s = \sqrt{s^2} = \sqrt{11.5} \approx 3.391$$

Answer:

Sample mean: 6.5 Sample variance: 11.5 Sample standard deviation: 3.391