Question

A car manufacturer takes an average of 17.5 hours to construct a car. This includes time for stamping, welding, painting, assembly and inspections. Construction times vary with a standard deviation of 30 minutes and these times follow a normal distribution. Show your working, including an appropriate diagram. (You can simply paste a photo of the diagram into your solution).
Question: What is the probability that a randomly selected car manufactured at this plant takes between 18 and 19 hours to construct?

Answer 1

Answer on Question #54323 – Math – Statistics and Probability

A car manufacturer takes an average of 17.5 hours to construct a car. This includes time for stamping, welding, painting, assembly and inspections. Construction times vary with a standard deviation of 30 minutes and these times follow a normal distribution.

Question: What is the probability that a randomly selected car manufactured at this plant takes between 18 and 19 hours to construct?

Solution

Given times follow a normal distribution with the average of $E(X) = 17.5$ hours and the standard deviation of $sd(X) = \frac{30}{60} = 0.5$ hour, the probability that a randomly selected car manufactured at this plant takes between 18 and 19 hours to construct is

$$P(18 < X < 19) = P\left(\frac{18 - E(X)}{sd(X)} < \frac{X - E(X)}{sd(X)} < \frac{19 - E(X)}{sd(X)}\right) = P\left(\frac{18 - 17.5}{0.5} < Z < \frac{19 - 17.5}{0.5}\right) = \\ = P(1 < Z < 3) = P(Z < 3) - P(Z < 1),$$

where $X \sim N\left(17.5; \left(\frac{30}{60}\right)^2\right)$, $Z \sim N(0; 1)$ are two random normally distributed variables.

From z-table we know

$$P(Z < 1) = 0.8413; \quad P(Z < 3) = 0.9987.$$

Thus,

$$P(18 < X < 19) = 0.9987 - 0.8413 = 0.1574.$$

Answer: 0.1574.

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