Question

Samples of 400 printed circuit boards were selected from each of two production lines \(A\) and \(B\). Line A produced 40 defectives, and line B produced 80 defectives. Estimate the difference in the actual fractions of defectives for the two lines with a confidence coefficient of \(.90 .\)

Short answer

Answer: The estimated difference between the actual fractions of defective printed circuit boards for production lines A and B with a confidence coefficient of 0.90 is between -0.15 and -0.05.

Step-by-step solution

Calculate the sample proportions of defectives in both lines

To calculate the sample proportions, we divide the number of defective boards by the total number of boards in each sample. \(p_A = \frac{40}{400} = 0.10\) and \(p_B = \frac{80}{400} = 0.20\)

Calculate the difference in sample proportions

Next, we need to calculate the difference in the sample proportions. \(p_{diff} = p_A - p_B = 0.10 - 0.20 = -0.10\)

Calculate the standard error of the difference

The standard error of the difference can be calculated using the formula: \(SE = \sqrt{\frac{p_A(1-p_A)}{n_A} + \frac{p_B(1-p_B)}{n_B}} = \sqrt{\frac{0.10(1-0.10)}{400} + \frac{0.20(1-0.20)}{400}} ≈ 0.03055\)

Find the critical value (z-score) corresponding to the given confidence level

We need to find the z-score that corresponds to the 0.90 confidence level. Since we want to estimate a difference, we will use a two-tailed test. The area in each tail is \((1 - 0.90) / 2 = 0.05\). Therefore, we need the z-score corresponding to an area of 0.95 (0.90 + 0.05) in the standard normal table. The z-score is approximately 1.645.

Calculate the confidence interval

Now, we can calculate the confidence interval using the formula: (p_A - p_B) ± z * SE. The confidence interval is given by: \(p_{diff} \pm z \cdot SE = -0.10 \pm 1.645 \cdot 0.03055 ≈ (-0.15 , -0.05)\) Thus, we can estimate with 90% confidence that the difference in the actual fractions of defectives between production lines A and B is between -0.15 and -0.05.