We appreciate your visit to According to a survey by the National Center for Health Statistics the heights of adult men in the U S are normally distributed with a. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
The probability that adult men are at most 59 inches or at least 74 inches tall is approximately 0.015 or 1.5%.
To find the probability that adult men are at most 59 inches or at least 74 inches tall, we need to calculate the area under the normal distribution curve for these two scenarios separately and then add them.
First, let's calculate the z-scores for 59 inches and 74 inches using the formula:
[tex]z=\frac{X-\mu}{\sigma}[/tex]
Where:
X is the value (height in inches),
μ is the mean (68 inches),
σ is the standard deviation (2.75 inches).
For 59 inches:
[tex]z_1=\frac{59-68}{2.75}=-\frac{9}{2.75} \approx-3.27[/tex]
For 74 inches:
[tex]z_2=\frac{74-68}{2.75}=\frac{6}{2.75} \approx 2.18[/tex]
Now, we need to find the probability corresponding to these z-scores using a standard normal distribution table or calculator.
Probability for
z≤−3.27
Probability for
z≥2.18
Then, we add these two probabilities together to get the total probability.
Let's calculate each probability:
Probability for z≤−3.27:
Using a standard normal distribution table or calculator, the probability for z≤−3.27 is very close to 0.001.
Probability for z≥2.18:
Using a standard normal distribution table or calculator, the probability for z≥2.18 is approximately 0.014.
Now, let's add these probabilities together:
P(X≤59 or X≥74)≈0.001+0.014=0.015
So, the probability that adult men are at most 59 inches or at least 74 inches tall is approximately 0.015 or 1.5%.
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Rewritten by : Barada
Using the properties of a normal distribution, the probability that an adult male in the U.S. is at most 59 inches or at least 74 inches can be found by calculating the z-scores for these heights and summing the probabilities associated with these z-scores using a standard normal distribution table or software.
The student is asking about the probability that the heights of adult men in the U.S., which are normally distributed, fall outside of a specified range. To find this probability, we'll need to use the properties of a normal distribution with the given mean of 68 inches and a standard deviation of 2.75 inches.
First, we'll calculate the z-scores for the heights 59 inches (at most) and 74 inches (at least) using the formula:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
For 59 inches:
Z1 = (59 - 68) ÷ 2.75 = -3.27
For 74 inches:
Z2 = (74 - 68) ÷ 2.75 = 2.18
Using a standard normal distribution table or software, we can find the probability associated with each of these z-scores. The probability that a man is at most 59 inches tall (Z1) will be the area to the left of Z1 in the normal distribution, and the probability that a man is at least 74 inches tall (Z2) will be the area to the right of Z2.
The total probability sought is the sum of these two probabilities:
Total Probability = P(Z ≤ 3.27) + P(Z ≥ 2.18)
To answer the student's question, these probabilities would be looked up or calculated using the z-scores, and if we assume typical normal probability values for these z-scores, we could say that:
P(Z ≤ -3.27) is very small, often too small to be observed in standard tables, and P(Z ≥ 2.18) might be around 1.5%.
