Final answer:
To determine the 90% confidence interval for the difference between failure rates of two types of microchips, the formula for the confidence interval is applied, and calculations result in a 90% confidence interval ranging from -0.0046 to 0.0856, after rounding to four decimal places.
Step-by-step explanation:
To find the 90% confidence interval for the difference between the proportions of failures for chips manufactured by the two processes, we use the following formula for the confidence interval of the difference between two proportions:
CI = (p1 - p2) ± Z*(√(p1*(1-p1)/n1 + p2*(1-p2)/n2))
Where:
- p1 = proportion of failures in the first group = 125/640
- p2 = proportion of failures in the second group = 130/840
- n1 = size of the first sample = 640
- n2 = size of the second sample = 840
- Z* = Z value for the chosen confidence level (90% confidence level corresponds to Z* approximately 1.645)
First, we calculate the sample proportions:
- p1 = 125/640 = 0.1953
- p2 = 130/840 = 0.1548
Then, we calculate the standard error (SE):
SE = √((0.1953*(1-0.1953)/640) + (0.1548*(1-0.1548)/840))
SE = √(0.000475 + 0.000275) = √(0.000750) = 0.0274
Now, we find the margin of error (MOE) using the Z* value:
MOE = 1.645 * 0.0274 = 0.0451
Finally, we calculate the confidence interval (CI):
CI = (0.1953 - 0.1548) ± 0.0451
CI = (0.0405 ± 0.0451)
CI = (-0.0046, 0.0856)
After rounding to four decimal places, our 90% confidence interval for the difference in the failure rates is (-0.0046, 0.0856).