Question

On average, the parts from a supplier have a mean of 31.8 inches and a standard deviation of 2.4 inches. Find the probability that a randomly selected part from this supplier will have a value between 27.0 and 36.6 inches. Is this consistent with the Empirical Rule of 68%-95%-99.7%?

Answer 1

Answer:

The probability is P=0.9545.

The empirical rule is consistent, as it would have estimate a probability of 95%.

Step-by-step explanation:

We have a random variable with mean 31.8 in. and standard deviation of 2.4 in.

We have to find the probability that a randomly selected part from this supplier will have a value between 27.0 and 36.6 in.

If we assumed a normal-like distribution, we can calculate the z-score for both values:

$z_1=\dfrac{X_1-\mu}{\sigma}=\dfrac{27-31.8}{2.4}=\dfrac{-4.8}{2.4}=-2\\\\\\z_2=\dfrac{X_2-\mu}{\sigma}=\dfrac{36.6-31.8}{2.4}=\dfrac{4.8}{2.4}=2$

That can be interpreted as an interval that is 2 standard deviations wide, centered on the mean.

This interval has a probability of:

$P(27.0$

If we apply the empirical rule, with 2 standard deviations from the mean, we would have estimate a probability of 95%, which is accurate.