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Two random samples were selected independently from populations having normal distributions. The statistics given below were extracted from the samples. Complete parts a through c. xˉ1=47.9xˉ2=37. 8 c. If σ1=σ2,s1=5, and s2=2, and the sample sizes are n1=20 and n2=20, construct a 90% confidence interval for the difference between the two population means. The confidence interval is ≤(μ1−μ2)≤ (Round to two decimal places as needed.)

User Jortx
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Final answer:

  • a. The confidence interval for the difference between the two population means is (8.899, 11.301).
  • b.The confidence interval for the difference between the two population means is (-5.599, 25.799).
  • c.The 99% confidence interval for the difference between the two population means is approximately 6.82 to 13.38.

Step-by-step explanation:

a. To construct a 99% confidence interval for the difference between the two population means when σ₁ = 3, σ₂ = 2, n₁ = 60, and n₂ = 60, we can use the formula:

CI = (x₁ - x₂) ± Z * √((σ₁² / n1) + (σ₂² / n₂))

where x1 and x₂ are the sample means, σ1 and σ₂ are the population standard deviations, n1 and n₂ are the sample sizes, and Z is the z-value corresponding to the desired confidence level.

Substituting the given values into the formula:

CI = (47.9 - 37.8) ± 2.58 * √((3² / 60) + (2² / 60))

CI = 10.1 ± 2.58 * √(0.15 + 0.0667)

CI = 10.1 ± 2.58 * √(0.2167)

CI = 10.1 ± 2.58 * 0.4656

CI = 10.1 ± 1.201

The confidence interval for the difference between the two population means is (8.899, 11.301).

b. When σ₁ = σ₂, we can use the formula:

CI = (x₁ - x₂) ± t * √((s₁² / n₁) + (s₂² / n₂))

where x₁ and x₂ are the sample means, s₁ and s₂ are the sample standard deviations, n₁ and n₂ are the sample sizes, and t is the t-value corresponding to the desired confidence level.

Since we want a 99% confidence interval and the sample sizes are small (n₁ = 10, n₂ = 10), we need to use the t-distribution and the t-value will be approximately 3.250 (from the t-table).

Substituting the given values into the formula:

CI = (47.9 - 37.8) ± 3.250 * √((3² / 10) + (2²/ 10))

CI = 10.1 ± 3.250 *√(0.9 + 0.4)

CI = 10.1 ± 3.250 *√(1.3)

CI = 10.1 ± 3.250 * 1.1402

CI = 10.1 ± 3.699

The confidence interval for the difference between the two population means is (-5.599, 25.799).

c) To construct a 99% confidence interval for the difference between the two population means when σ1 ≠ σ₂, s₁ = 3, s₂ = 2, n₁= 10, and n₂ = 10, we can use the formula:

CI = (x₁ - x₂) ± t * √((s₁² / n₁) + (s₂²/ n₂))

where x₁ and x₂ are the sample means, s₁ and s₂ are the sample standard deviations, n₁ and n₂ are the sample sizes, and t is the t-value corresponding to the desired confidence level.

First, let's calculate the difference between the sample means:

x₁ - x₂ = 47.9 - 37.8 = 10.1

Next, let's determine the t-value for a 99% confidence level with (n₁+ n₂ - 2) degrees of freedom. Since both sample sizes are 10, the degrees of freedom will be 10 + 10 - 2 = 18. Using a t-table or a statistical software, the t-value for a 99% confidence level and 18 degrees of freedom is approximately 2.878.

Now, let's calculate the standard error of the difference:

√((s₁² / n1) + (s₂² / n₂)) =√((3² / 10) + (2²/ 10)) =√(0.9 + 0.4) = √(1.3) ≈ 1.14

Finally, we can construct the 99% confidence interval:

CI = (10.1) ± (2.878 * 1.14) = 10.1 ± 3.28

Therefore, the 99% confidence interval for the difference between the two population means is approximately 6.82 to 13.38...

Your question is incomplete, but most probably the full question was:

Two random samples were selected independently from populations having normal distributions. The statistics given below were extracted from the samples. Complete parts a through c.

x=47.9

x2=37.8.

a. If σ=3 and σ=2 and the sample sizes are n=60 and n=60​, construct a 99​% confidence interval for the difference between the two population means.

The confidence interval is

≤μ−μ

​(Round to two decimal places as​ needed.)

b. If σ​, s=3​, and s=2​, and the sample sizes are n=10 and n=10​, construct a 99​% confidence interval for the difference between the two population means.

The confidence interval is?

nothing≤μ1−μ≤nothing.

​(Round to two decimal places as​ needed.)

c. If σ≠σ, s1=3​, and s=2​, and the sample sizes are n=10 and n=10​, construct a 99​% confidence interval for the difference between the two population means.

The confidence interval is?

nothing≤μ−μ≤nothing.

​(Round to two decimal places as​ needed.)

Two random samples were selected independently from populations having normal distributions-example-1
User Chris Melinn
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