Based on data from the National Health and Nutrition Examination Survey assume that the weights of men are normally distributed with a mean of 170 lbs. and a standard deviation of 30lbs.
a. Find the probability that a man randomly selected from the above population will have a weight equal to orless than 207.5 lbs.
b. Find the probability thata man randomly selected from the above population will have weight equal to or greater than 125 lbs.
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c. Find the probability that a man is randomly selected from the above population will have a weight between125 lbs and 215 lbs.
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Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
To find the probabilities in these scenarios, we can use the information provided: the mean weight of men is 170 lbs and the standard deviation is 30 lbs. We can utilize the z-score formula to standardize the values and then find the corresponding probabilities in the standard normal distribution.
The z-score formula is:
z = (x - μ) / σ
Where:
x is the given value,
μ is the mean of the population, and
σ is the standard deviation.
a. Find the probability that a man randomly selected from the above population will have a weight equal to or less than 207.5 lbs.
To find this probability, we need to find the z-score for the weight of 207.5 lbs and then find the area under the standard normal curve corresponding to that z-score.
z = (207.5 - 170) / 30 = 1.25
Now, we can look up the area under the standard normal curve for a z-score of 1.25. Using a standard normal distribution table or a calculator, we find the corresponding area to be approximately 0.8944.
Therefore, the probability that a man randomly selected from the population will have a weight equal to or less than 207.5 lbs is 0.8944 or 89.44%.
b. Find the probability that a man randomly selected from the above population will have a weight equal to or greater than 125 lbs.
Similarly, we need to find the z-score for the weight of 125 lbs and then find the corresponding area under the standard normal curve.
z = (125 - 170) / 30 = -1.5
Looking up the area for a z-score of -1.5 in the standard normal distribution table or using a calculator, we find the corresponding area to be approximately 0.0668.
Therefore, the probability that a man randomly selected from the population will have a weight equal to or greater than 125 lbs is 0.0668 or 6.68%.
c. Find the probability that a man randomly selected from the above population will have a weight between 125 lbs and 215 lbs.
To find this probability, we need to find the individual probabilities for each weight separately and then subtract the probability of the lower end value from the probability of the upper end value.
For the lower end value:
z(lower) = (125 - 170) / 30 = -1.5
The area corresponding to a z-score of -1.5 is approximately 0.0668.
For the upper end value:
z(upper) = (215 - 170) / 30 = 1.5
The area corresponding to a z-score of 1.5 is approximately 0.9332.
Therefore, the probability that a man randomly selected from the population will have a weight between 125 lbs and 215 lbs is 0.9332 - 0.0668 = 0.8664 or 86.64%.
Please note that these probabilities are approximate values and used for illustration purposes. In practice, more precise calculations may be necessary using statistical software or calculators.