Two Proportion Z-Interval Practice

Question

A large company has two shifts—a day shift and a night shift. Parts produced by the two shifts must meet the same specifications. The manager of the company believes that there is a difference in the proportions of parts produced within specifications by the two shifts. To investigate this belief, random samples of parts that were produced on each of these shifts were selected. For the day shift, 188 of its 200 selected parts met specifications. For the night shift, 180 of its 200 selected parts met specifications.

A. Use a 96 percent confidence interval to estimate the difference in the proportions of parts produced within specifications by the two shifts.

b> Based only on this confidence interval, do you think that the difference in the proportions of parts produced within specifications by the two shifts is significantly different from 0 ? Justify your answer.

Answer 1

Based on the given information let:

1. Day Shift be Group 1,
2. Night Shift be Group 2, because of the 2 shifts.
3. G₁ = 188 / 200 = 0.94 → Day shift
4. G₂ = 180 / 200 = 0.90 → Night shift

We can calculate the difference in sample proportions, which will be:

1. G₁ - G₂ = 0.94 - 0.90 = 0.04

Part A: 96% Confidence interval for G₁ - G₂

Step 1: Standard Error

1. SE = √(G₁(1 - G₁)) / n₁ + (G₂(1 - G₂)) / n₂
2. SE = √(0.94(0.06)) / 200 + (0.90(0.10))/200
3. SE = √0.000282 + 0.00045
4. SE = √0.000732 ≈ 0.0271

Step 2: Critical Value

1. For a 96% confidence level: z^* ≈ 2.05

Step 3: Margin of error

1. ME = 2.05 (0.0271) ≈ 0.0556

Step 4: Confidence interval

1. 0.04 ± 0.0556

→ (-0.0156, 0.0956)

That means that the 96% confidence interval would be: (-0.016, 0.096)

Part B: is the difference significantlly different from 0? α

1. No, because the confidence interval contains 0. That meansthat the difference of - is a plausible value for G₁-G₂