The following are independent samples from two normal populations , both of which have the same standard deviation ó. 16,17,19,20,18 and 3,4,8.use them to estimate ó.
To estimate the common standard deviation (σ) of the two populations based on the given independent samples, you can use the concept of pooled variance.
The steps to estimate σ are as follows:
Step 1: Calculate the sample variances for each sample.
- Sample 1 (first set of observations): Calculate the sample variance (s₁²) using the formula: s₁² = (∑(x - x̄)²) / (n - 1), where x is the individual observation, x̄ is the sample mean, and n is the sample size.
- Sample 2 (second set of observations): Calculate the sample variance (s₂²) using the same formula mentioned above.
For the first sample (16, 17, 19, 20, 18):
- Calculate the sample mean (x̄₁) by adding all the observations and dividing by the sample size.
x̄₁ = (16 + 17 + 19 + 20 + 18) / 5 = 90 / 5 = 18
- Calculate the sum of squared differences, (∑(x - x̄)²), by subtracting x̄₁ from each observation, squaring the result, and adding them all up.
(∑(x - x̄)²) = (16 - 18)² + (17 - 18)² + (19 - 18)² + (20 - 18)² + (18 - 18)²
= 4 + 1 + 1 + 4 + 0
= 10
- Divide (∑(x - x̄)²) by (n - 1) to get the sample variance (s₁²).
s₁² = 10 / 4 = 2.5
For the second sample (3, 4, 8):
- Calculate the sample mean (x̄₂) in the same way as before.
x̄₂ = (3 + 4 + 8) / 3 = 15 / 3 = 5
- Calculate (∑(x - x̄)²).
(∑(x - x̄)²) = (3 - 5)² + (4 - 5)² + (8 - 5)²
= 4 + 1 + 9
= 14
- Divide (∑(x - x̄)²) by (n - 1) to get the sample variance (s₂²).
s₂² = 14 / 2 = 7
Step 2: Calculate the pooled variance.
- Combine the variances from both samples to get the pooled variance (s_pooled²) using the formula: s_pooled² = ((n₁ - 1) * s₁² + (n₂ - 1) * s₂²) / (n₁ + n₂ - 2), where n₁ and n₂ are the sample sizes.
- Substitute the values into the formula and calculate the result.
s_pooled² = ((5 - 1) * 2.5 + (3 - 1) * 7) / (5 + 3 - 2)
= (4 * 2.5 + 2 * 7) / 6
= (10 + 14) / 6
= 24 / 6
= 4
Step 3: Calculate the estimate of the common standard deviation.
- Take the square root of the pooled variance to get the estimate of the common standard deviation.
σ = √(s_pooled²) = √4 = 2
Therefore, the estimate of the common standard deviation (σ) based on the given samples is 2.