a companys ceo wanted to estimate the percentage of defective product per shipment. in a sample contianing 600 products he found 45 defective product. a. find 99% confidence interval for the true porporton of defective product show calcultions and explain process used to obtain interval. (b) interpret thic confidence interval and write a sentence to explain it
To find the 99% confidence interval for the true proportion of defective products in the shipment, we can use the formula:
CI = p̂ ± z * √(p̂(1-p̂)/n)
where CI is the confidence interval, p̂ is the sample proportion, z is the z-score corresponding to the desired confidence level, √ is the square root, and n is the sample size.
(a) Let's calculate the confidence interval step by step:
Step 1: Calculate the sample proportion (p̂):
p̂ = number of defective products / total sample size
= 45 / 600
= 0.075
Step 2: Find the z-score corresponding to a 99% confidence level.
For a 99% confidence level, we need to find the z-score that leaves 0.5% in each tail of the standard normal distribution. The z-score for a 99% confidence level is approximately 2.576.
Step 3: Calculate the standard error (SE):
SE = √(p̂(1-p̂)/n)
= √(0.075(1-0.075)/600)
≈ 0.0118
Step 4: Calculate the margin of error (MOE):
MOE = z * SE
= 2.576 * 0.0118
≈ 0.0304
Step 5: Calculate the lower and upper limits for the confidence interval:
Lower limit = p̂ - MOE
= 0.075 - 0.0304
= 0.0446
Upper limit = p̂ + MOE
= 0.075 + 0.0304
= 0.1054
Therefore, the 99% confidence interval for the true proportion of defective products per shipment is approximately 0.0446 to 0.1054.
(b) To interpret this confidence interval, we can say that we are 99% confident that the true proportion of defective products in the entire shipment lies between 4.46% and 10.54%.
To find the 99% confidence interval for the true proportion of defective products, we can use the formula:
CI = p̂ ± Z * √(p̂(1-p̂)/n)
Where:
- p̂ is the sample proportion of defective products
- Z is the Z-score corresponding to the desired confidence level
- n is the sample size
Let's calculate it step-by-step:
(a) Calculation:
1. Calculate the sample proportion of defective products:
p̂ = (number of defective products) / (total sample size)
p̂ = 45/600 = 0.075
2. Calculate the Z-score corresponding to the 99% confidence level. Since we want a two-tailed test, we divide the desired significance level by 2 and look up the critical Z-values using a Z-table or calculator. For a 99% confidence level, Z = 2.576.
3. Calculate the standard error:
SE = √(p̂(1-p̂)/n)
SE = √((0.075)(1-0.075)/600) ≈ 0.0128
4. Calculate the margin of error:
MOE = Z * SE
MOE = 2.576 * 0.0128 ≈ 0.0330
5. Calculate the lower and upper bounds of the confidence interval:
Lower bound = p̂ - MOE
Lower bound = 0.075 - 0.0330 ≈ 0.0420
Upper bound = p̂ + MOE
Upper bound = 0.075 + 0.0330 ≈ 0.1080
Therefore, the 99% confidence interval for the true proportion of defective products is approximately 0.0420 to 0.1080.
(b) Interpretation:
We are 99% confident that the true proportion of defective products in the company's shipments falls within the interval of 0.0420 to 0.1080. This means that based on the given sample, we can estimate with 99% confidence that the percentage of defective products in the company's shipments lies between 4.20% and 10.80%.