Question

Let X denote the number of flaws along a 100-m reel of magnetic tape. Suppose X has approximately a normal distribution with μ= 25 and σ= 5. Use the continuity correlation to calculate the probability that the number of flaws is

a. Between 20 and 30, inclusive.
b. At most 30. Less than 30.

Answer 1

for part a, I started with
P=(20≤X≤30) = P [((20−.5−25)/5) ≤ ((x−μ)/σ) ≤ ((30+.5−25)/5)]

Answer 2

You need to find the Z-scores that correspond to those particular values for the number of flaws. Remember that
So for part a, the z scores are
\[Z_{30} = \frac{30 - 25}{5} = 1\]
\[Z_{20} = \frac{20-25}{5} = -1\]
If you look those values up in a Z-score chart and subtract properly, you find the area between those two values to be
\[P(20 \le x \le 30) = 0.8413 - 0.1587 = 0.6826\]
For part b, you use the same procedure...
\[P(x\le 30) = 0.8413\]
\[P(30 \le x) = 1 - P(x \le 30) = 1-0.8413 = 0.1587\]

Z-score charts generally give you all the area under the curve from negative infinity up to the value in question.

Answer 3

Thanks!