The height of an adult male is known to be normally distributed with a mean of 69 inches and a standard deviation of 2.5 inches. The height of the doorway is 74 inches. What proportion of adult males will not fit under the door?
.9772
To find the proportion of adult males who will not fit under the door, we need to calculate the area under the normal distribution curve above the doorway height of 74 inches.
First, we need to standardize the door height using the formula:
Z = (X - μ) / σ
where X is the doorway height, μ is the mean height, and σ is the standard deviation.
Z = (74 - 69) / 2.5 = 2
Next, we need to find the proportion of the area above Z = 2. We can use a standard normal distribution table or a calculator to find this value.
Using a standard normal distribution table, the area to the right of Z = 2 is approximately 0.0228.
Therefore, the proportion of adult males who will not fit under the door is approximately 0.0228 or 2.28%.
To find the proportion of adult males who will not fit under the door, we need to find the area under the curve of the normal distribution that represents heights higher than the doorway height.
Since we know that the height of adult males is normally distributed with a mean (μ) of 69 inches and a standard deviation (σ) of 2.5 inches, we can use this information to calculate the z-score for the doorway height.
A z-score measures how many standard deviations an observation is from the mean. It can be calculated using the formula:
z = (x - μ) / σ
where x is the observation (doorway height in this case), μ is the mean, and σ is the standard deviation.
Let's calculate the z-score for the doorway height:
z = (74 - 69) / 2.5
z = 5 / 2.5
z = 2
Now that we have the z-score, we can use a standard normal distribution table or a calculator that provides the cumulative probability to find the proportion of heights higher than the doorway height.
Looking up the z-score of 2 in a standard normal distribution table or using a calculator, we find that the cumulative probability is approximately 0.9772.
Since we are interested in the proportion of heights higher than the doorway height, we subtract the cumulative probability from 1:
Proportion = 1 - 0.9772
Proportion ≈ 0.0228
Therefore, approximately 0.0228 or 2.28% of adult males will not fit under the door.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.