Statistics question

Question

2) The number of auto parts that a company produces follows a normal distribution. The company has two independent production lines: Line 1 produces an average of 75 parts per day with a standard deviation of 20. Line 2 produces an average of 65 parts per day with a standard deviation of 21. Line 1 and Line 2 are independent. The probability that Line 1 produces more parts than Line 2 in any single day is closest to A) 0.3413 B) 0.1587 C) 0.6331 D) 0.7794 E) 0.5

Answer 1

To answer this, we need to understand the distribution of Line1 - Line2 and get P(Line1 > Line2)

The difference of independent normals is normal.

The mean of Line1 - Line2 = 75 - 65 = 10

If Line1 and Line2 are independent, the variance of the difference is the sum of the variances.

Thus, the standard deviation of Line1 - Line2 = √(21² + 20²) = 29

Thus, the mean is 10 and the standard deviation is 29, so P(Line 1 - Line 2) > 0 =

Φ(10/29) = from Excel, norm.s.dist(10/29,1) = .6349 (FYI, the 1 is to indicate the cumulative value rather than density of the function at 10/29)

We could note that, of the listed answers, this is closest to .6331, or round 10/29 to .34 and see that norm.s.dist(.34,1) = .6331. (This is how problems were solved when we relied on tables in books.)