Answer:
a) P(a > 80) = 0.323
b) The 95% confidence interval = (73.40, 80.60)
c) The 95% confidence interval expresses that the mean of the distribution can always be found in the given range, with a 95% confidence level.
Explanation:
X ~ (45, 4)
Y ~ (32, 2.5)
(X+Y) ~ (77, 6.5)
Let a = (X+Y)
a) Probability that the time required to complete both of those steps will exceed 80 min = P(a > 80)
This is a normal distribution problem
We then standardize 80 min time
The standardized score for any value is the value minus the mean then divided by the standard deviation.
z = (a - μ)/σ = (80 - 77)/6.5 = 0.46
To determine the probability the time required to complete both of those steps will exceed 80 min = P(a > 80) = P(z > 0.46)
We'll use data from the normal probability table for these probabilities
P(a > 80) = P(z > 0.46) = 1 - P(z ≤ 0.46) = 1 - 0.677 = 0.323
b) 95% confidence interval for the expected total time required to produce one flow meter.
We need to obtain the margin of error
Margin of error = (critical value) × (standard error of the sample)
Critical value for a 95% confidence interval = critical value for a significance level of 5% = t(15-1, 0.05/2) = 2.145 (using the t-score since information on the population mean and standard deviation isn't known)
Standard error for the sample of sum of times = (standard deviation of the sum of times)/√n = (6.5/√15) = 1.678
Margin of error = 2.145 × 1.678 = 3.60
Limits of the confidence interval = (Sample mean ± margin of error)
Lower limit of the confidence interval = (Sample mean - margin of error) = 77 - 3.60 = 73.40
Upper limit of the confidence interval = 77 + 3.60 = 80.60
The confidence interval = (73.40, 80.60)