A certain company produces vinyl for records in three grades (A, B, and C). Over time, grade A vinyl has a 3% defective rate, grade B vinyl has a 5% defective rate, and grade C vinyl has a 10% defective rate. Since grade A vinyl is less likely to be defective, the company can charge more money than it can charge for the other two grades. The company can also charge more money for grade B vinyl than grade C vinyl.
Recently, the company found a batch of unlabeled vinyl in a warehouse. They could determine the vinyl was from the same batch. The company randomly sampled 150 discs and found 8 defective discs.
(a) Construct and interpret a 90% confidence interval for the proportion of defective discs in the batch.
(b) Does the interval calculated in part (a) clearly allow the company to determine the grade of the vinyl?
To construct and interpret a confidence interval for the proportion of defective discs in the batch, we can use the formula for confidence interval of a proportion. Let's go through the steps:
Step 1: Identify the relevant values
The sample size is 150 discs (n = 150) and the number of defective discs in the sample is 8 (x = 8). We also need the confidence level, which is given as 90%.
Step 2: Calculate the sample proportion
The sample proportion (p̂) can be calculated by dividing the number of defective discs by the sample size:
p̂ = x / n = 8 / 150 = 0.0533
Step 3: Calculate the standard error
The standard error (SE) can be calculated using the following formula:
SE = √((p̂ * (1 - p̂)) / n)
SE = √((0.0533 * (1 - 0.0533)) / 150) ≈ 0.0221
Step 4: Determine the z-score
To find the z-score corresponding to a 90% confidence level, we use the z-table or calculator. In this case, the z-score is approximately 1.645.
Step 5: Calculate the margin of error
The margin of error (ME) can be calculated by multiplying the standard error by the z-score:
ME = z * SE ≈ 1.645 * 0.0221 ≈ 0.0364
Step 6: Calculate the confidence interval
The confidence interval can be calculated by subtracting and adding the margin of error from the sample proportion:
Confidence interval = p̂ ± ME
Confidence interval = 0.0533 ± 0.0364
Confidence interval ≈ (0.0169, 0.0897)
Interpretation (a):
We are 90% confident that the true proportion of defective discs in the batch is between 0.0169 and 0.0897.
Moving on to part (b):
Based on the confidence interval calculated in part (a), it does not clearly allow the company to determine the grade of the vinyl. The confidence interval provides a range of values within which we estimate the true proportion of defective discs in the batch. It does not directly tell us the grade of the vinyl. To determine the grade of the vinyl with more certainty, further analysis or testing is needed.