Question

Forced expiratory volume (FEV) is an index of pulmonary function that measures the volume of air expelled after 1 second. FEV is influenced by age, sex and cigarette smoking. Assume that in 45- to 54-year-old nonsmoking men FEV is normally distributed with mean = 4.0 L and standard deviation = 0.5L.

But for 45- to 54-year-old smoking men FEV is normally distributed, with mean 3.5 L and standard deviation = 0.6 L.
(a) If an FEV of less than 2.5 L is regarded as showing some functional impairment (occasional breathlessness, inability to climb stairs, etc.), then what is the probability that a currently smoking man in the 45- to 54-age group has functional impairment?

Answer 1

To determine the probability of functional impairment in smoking men aged 45 to 54 with an FEV less than 2.5L, one calculates a z-score of -1.67 using the mean FEV of 3.5L and standard deviation of 0.6L. The corresponding probability from a standard normal distribution is approximately 4.75%, meaning there is a 4.75% chance of impairment.

Step-by-step explanation:

The student's question involves calculating the probability that a currently smoking man in the 45- to 54-year-old age group has a functional impairment, which is defined as having an FEV of less than 2.5L. To find this probability, we need to use the properties of the normal distribution that applies to the smoking men's FEV with a mean of 3.5 L and a standard deviation of 0.6 L.

First, we calculate the z-score, which represents how many standard deviations an element is from the mean. The z-score is calculated using the formula:

Z = (X - μ) / σ

Where X is the value of interest (2.5L), μ is the mean (3.5L), and σ is the standard deviation (0.6L). Plugging in the values, we get:

Z = (2.5L - 3.5L) / 0.6L = -1.67

Next, we look up the z-score in a standard normal distribution table or use a calculator capable of cumulative normal distribution functions to find the probability corresponding to a z-score of -1.67. The area to the left of this z-score gives us the probability that a smoking man has a FEV less than 2.5L, indicating functional impairment. Let's assume this area is P approximately equal to 0.0475, representing the probability, or 4.75% chance, that a smoking man in this age group will have an FEV less than 2.5L.