Question

1.    A population of hairs have diameters that has been shown to be normally distributed with a mean (µ) equal to 45 µm and standard deviation (σ) equal to 10 µm.   Using this information find the probability of a hair having:

 

a.    A diameter less than 30 µm.

b.    A diameter in the range 50-65 µm.

c.     A diameter within 3 µm of the mean.

Answer 1

To find the probabilities for the given diameter ranges, we will use the standard normal distribution table or a calculator to convert the values to standard z-scores and then look up the corresponding probabilities.

 

Given information:

Population mean (μ) = 45 pm

Standard deviation (σ) = 10 pm

 

a. Probability of a hair having a diameter less than 30 pm:

We need to find P(X < 30), where X is the diameter of a hair.

 

Z-score formula: Z = (X - μ) / σ

 

Z-score for X = 30 pm:

Z = (30 - 45) / 10 = -1.5

 

Using the standard normal distribution table or a calculator, we find that the probability corresponding to a Z-score of -1.5 is approximately 0.0668.

 

b. Probability of a hair having a diameter in the range 50-65 pm:

We need to find P(50 < X < 65), where X is the diameter of a hair.

 

Z-score for X = 50 pm:

Z = (50 - 45) / 10 = 0.5

 

Z-score for X = 65 pm:

Z = (65 - 45) / 10 = 2

 

Using the standard normal distribution table or a calculator, we find the probabilities corresponding to Z-scores of 0.5 and 2:

P(Z < 0.5) ≈ 0.6915

P(Z < 2) ≈ 0.9772

 

To find the probability for the range 50-65 pm, we subtract the lower probability from the higher probability:

P(50 < X < 65) = P(Z < 2) - P(Z < 0.5) ≈ 0.9772 - 0.6915 ≈ 0.2857

 

c. Probability of a hair having a diameter within 3 pm of the mean:

We need to find P(42 < X < 48), where X is the diameter of a hair.

 

Z-score for X = 42 pm:

Z = (42 - 45) / 10 = -0.3

 

Z-score for X = 48 pm:

Z = (48 - 45) / 10 = 0.3

 

Using the standard normal distribution table or a calculator, we find the probabilities corresponding to Z-scores of -0.3 and 0.3:

P(Z < -0.3) ≈ 0.3821

P(Z < 0.3) ≈ 0.6179

 

To find the probability for the range within 3 pm of the mean, we subtract the lower probability from the higher probability:

P(42 < X < 48) = P(Z < 0.3) - P(Z < -0.3) ≈ 0.6179 - 0.3821 ≈ 0.2358

 

So, the probabilities for the given diameter ranges are:

a. P(X < 30) ≈ 0.0668

b. P(50 < X < 65) ≈ 0.2857

c. P(42 < X < 48) ≈ 0.2358