Question

Diameters of a country's currency coins are normally distributed with a mean of 2 cm and standard deviation of 0.5 cm.If Aiden has a coin that is at the 40th percentile, what diameter does Aiden's coin measure to? Round your answer to onedecimal place. Do not include units in your answer.

Answer 1

The 40 percentile means that we are looking for the diameter of the coin (X) that separates the bottom 40% of the distribution from the top 60%.

Since we have to work with the standard normal distribution we have:

$P(Z\le z)=0.4$

Now we use the given table to determine the value of "z" that is closest to the probability of 0.4.

The way to read the table is the following. The first column tells you the first decimal digit of the z-value and the first row tells the second digit.

From the table we see that the closest number to the probability of 40% or 0.4 is 0.4013, this means that the z-value is z = -0.25. Now using the z-value formula:

$z=\frac{X-\mu}{\sigma}$

Since X = d, we get:

$z=\frac{d-\mu}{\sigma}$

Solving for "d" first by multiplying both sides by sigma:

$z\sigma=d-\mu$

Adding the mean to both sides:

$z\sigma+\mu=d$

Replacing the given values:

$(-0.25)(0.5)+2=d$

Solving the operations:

$1.875=d$

Therefore, the value of "d" is 1.875 cm. Rounded would be 1.9 cm.