Solved Examples on Point Estimation

Example 1: Calculate the sample mean for the following data set: {12, 15, 18, 21, 24}.

Solution:

Step 1: Add up all the values: 12 + 15 + 18 + 21 + 24 = 90.

Step 2: Divide the sum by the number of observations: 90 / 5 = 18.

Therefore, the sample mean is 18.

Example 2: Find the sample variance for the data set: {5, 8, 10, 12, 15}.

Solution:

Step 1: Calculate the sample mean using the same process as in the previous question. Mean = (5 + 8 + 10 + 12 + 15) / 5 = 50 / 5 = 10.

Step 2: Calculate the squared differences between each observation and the mean: (5 – 10)² + (8 – 10)² + (10 – 10)² + (12 – 10)² + (15 – 10)² = 25 + 4 + 0 + 4 + 25 = 58.

Step 3: Divide the sum of squared differences by the number of observations minus 1: 58 / (5 – 1) = 58 / 4 = 14.5.

Therefore, the sample variance is 14.5.

Example 3: Determine the sample proportion of successes if out of 50 trials, 30 were successful.

Solution:

Divide the number of successful trials by the total number of trials: 30 / 50 = 0.6.

Therefore, the sample proportion of successes is 0.6 or 60%.

Example 4: Calculate the sample median for the following data set A: {10, 15, 18, 20, 22}.

Solution:

Since, data set has an odd number of observations, the median is the middle value after arranging in ascending or descending order.

A = {10, 15, 18, 20, 22}

n = 5

It is already in descending order

Median = {(n + 1)/2}^th term

Median = (5 + 1)/2 = 3^rd term = 18

Example 5: Find the sample standard deviation for the data set: {3, 5, 7, 9, 11}.

Solution:

Step 1: Calculate the sample mean: (3 + 5 + 7 + 9 + 11) / 5 = 7.

Step 2: Calculate the squared differences between each observation and the mean: (3 – 7)² + (5 – 7)² + (7 – 7)² + (9 – 7)² + (11 – 7)² = 16 + 4 + 0 + 4 + 16 = 40.

Step 3: Divide the sum of squared differences by the number of observations minus 1: 40 / (5 – 1) = 40 / 4 = 10.

Step 4: Take the square root of the result: √10 ≈ 3.16.

Therefore, the sample standard deviation is approximately 3.16.

Point estimation is a fundamental concept in statistics providing a method for estimating population parameters based on sample data. In this article, we will discuss point estimation, its techniques and its significance in detail.