A normal distribution with mean 37.6 cc per minute and variance 4.6 cc per minute. Find the probabilities (a) at least 44.5 cc.

Answer 1

Step1. Conceptualize the question.

This is an Upper tail question, since they are asking for the probability of at least 44.5 [P(44.5cc >= Z)]. So we want to find the upper tail area under the curve when the value is at least 44.5 cc per minute. However, the probabilities under the curve going to the right approach infinity, which means we need to modify this inequality using the complement rule. So…

We need to

1. Convert 44.5 to a z score
2. Calculate the probability of that z score via appendix table.
3. Calculate the probability (aka, area under the curve) of that Z score.

Complement rule 

P(X>= Z)= 1- P(X<Z)

Step2. Note your Givens

Distribution: Normal

U = 37.6 

X = 44.5

σ^2 = 4.6 (Must take square root of this to find standard deviation)

σ= 2.144

Step3: Determine your Z value (to get this, we take our observation of interest, measurement of 44.5 cc, and convert it into a Z test value.

Using the formula

Z= (X - U) ÷ σ

We get

(44.5 - 37.6) ÷ 2.144 = 3.218 = Z value

Step 4. Statistical Appendix to derive area under the curve for P( Z<3.218) Remember, complement rule.

P( Z<3.218) ≈0.9990

Step 5. Plug into the complement rule.

P(44.5>= Z)= 1- P(44.5<Z) = 1-0.9990= .0001

Solution: The probability of finding a value of at least 44.5 CC is 0.0001, or .01%