Question

Use the information given to find the appropriate minimum sample sizes. (Round your answer up to the nearest whole number.) Estimating μ1​−μ2​ to within 0.16 with probability 0.90. Assume that the sample sizes will be equal and that σ12​=σ22​=29.3 n1​=n2​≥ You may need to use the appropriate appendix table to answer this question.

Answer 1

The minimum sample size for estimating μ₁ - μ₂ to within 0.16 with a probability of 0.90, assuming equal sample sizes and σ₁² = σ₂² = 29.3, is approximately 96.

Step-by-step explanation:

To calculate the minimum sample size needed for estimating the difference between two population means (μ₁ - μ₂) with a specified margin of error and confidence level, we can use the formula:

n = Z²(σ₁² + σ₂²) / E²

where:

(n) is the sample size for each group,

(Z) is the Z-score corresponding to the desired confidence level,

σ₁ and σ₂ are the standard deviations of the two populations, and

(E) is the desired margin of error.

In this case, the margin of error (E) is 0.16, and the confidence level is 0.90. The Z-score corresponding to a 90% confidence level is approximately 1.645. The standard deviation
`(\(\sigma\))` is given as 29.3 for both populations.

Plugging these values into the formula:

`\[n = \left((1.645^2 \cdot (29.3^2 + 29.3^2))/(0.16^2)\right)\]`

Calculating this expression yields the minimum sample size required for each group, and since equal sample sizes are assumed, the final answer is rounded up to the nearest whole number, resulting in a minimum sample size of approximately 96 for each group.

In summary, a sample size of 96 for each group is needed to estimate the difference between population means within 0.16 with a 90% confidence level, assuming equal sample sizes and equal population standard deviations of 29.3.