Question

A population of Douglas fir trees was shown to have diameters that follow an approximately Normal distribution, with a mean of 50 cm and a standard deviation of 12 cm. What is the maximum diameter a Douglas fir can have and still be in the bottom 8%? 37.5 cm ; 26.0 cm; 33.1 cm ; 62.4 cm

Answer 1

To find the maximum diameter in the bottom 8% of the population, calculate the z-score and use it to find the corresponding diameter.

Step-by-step explanation:

To find the maximum diameter a Douglas fir can have and still be in the bottom 8%, we need to find the z-score corresponding to the bottom 8% of the normal distribution. The z-score can be found using the cumulative standard normal distribution table or a calculator. In this case, the z-score is approximately -1.405.

Next, we can use the formula for z-scores to find the corresponding diameter:

diameter = mean + (z-score * standard deviation)

Substituting the values, we get:

diameter = 50 cm + (-1.405 * 12 cm) = 32.78 cm

Therefore, the maximum diameter a Douglas fir can have and still be in the bottom 8% is approximately 32.78 cm.

Answer 2

To find the maximum diameter a Douglas fir can have and still be in the bottom 8%, we use the normal distribution and the z-score formula. The maximum diameter is approximately 33.14 cm.

Step-by-step explanation:

To find the maximum diameter a Douglas fir can have and still be in the bottom 8%, we need to find the z-score corresponding to the bottom 8% of the normal distribution and then convert it back to a diameter value.

To find the z-score, we use the formula:

z = (x - mu) / sigma

where x is the diameter value, mu is the mean, and sigma is the standard deviation.

So, for the bottom 8%:

z = invNorm(0.08) ≈ -1.405

Now, we can solve for x:

-1.405 = (x - 50) / 12

cross-multiply on both sides

x - 50 = -1.405 * 12

solve for x

x - 50 = -16.86

x ≈ 33.14

Therefore, the maximum diameter a Douglas fir can have and still be in the bottom 8% is approximately 33.14 cm.