Bad Wheels Tire Company is interested in estimating the proportion of defective tires that it produces. The company tested a sample of 500 tires and found 16 to be defective. Find a 99% upper bound confidence interval for the true proportion of defective tires produced by the firm.
To find a confidence interval for the true proportion of defective tires produced by the company, we can use the formula for constructing a confidence interval for a proportion.
The formula is:
CI = p̂ ± Z * √(p̂ * (1 - p̂) / n)
Where:
- CI is the confidence interval
- p̂ is the sample proportion (in this case, the number of defective tires divided by the total number of tires)
- Z is the Z-score corresponding to the desired level of confidence (in this case, 99%)
- n is the sample size (in this case, 500)
First, calculate the sample proportion:
p̂ = 16/500 = 0.032
Next, determine the Z-score for a 99% confidence level. You can look up this value using a standard normal distribution table or use a statistical software.
For a two-tailed test, the Z-score for a 99% confidence level is approximately 2.576.
Now, substitute the values into the formula:
CI = 0.032 ± 2.576 * √(0.032 * (1 - 0.032) / 500)
Calculating the square root term: √(0.032 * (1 - 0.032) / 500) ≈ 0.010
CI = 0.032 ± 2.576 * 0.010
Finally, calculate the upper and lower bounds of the confidence interval:
Upper bound: 0.032 + (2.576 * 0.010) = 0.0576
Therefore, the 99% upper bound confidence interval for the true proportion of defective tires produced by Bad Wheels Tire Company is approximately 0.005 - 0.058.