The weights of a group of adult males are normally distributed with a mean of 180lb and standard deviation of 8lb. Find the probability that a randomly chosen male from this group weighs more than 190lb.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the probability that a randomly chosen male from this group weighs more than 190lb, you need to use the standard normal distribution table or a calculator that can compute the cumulative probability of a normal distribution.
Here's how you can calculate it step by step using the standard normal distribution table:
Step 1: Calculate the z-score.
The z-score represents the number of standard deviations an individual value is from the mean. In this case, you want to find the z-score for a weight of 190lb.
z = (x - μ) / σ
Where:
- x is the weight value (190lb),
- μ is the mean (180lb),
- σ is the standard deviation (8lb).
z = (190 - 180) / 8 = 10 / 8 = 1.25
Step 2: Use the standard normal distribution table.
Look up the z-score of 1.25 in the standard normal distribution table. The table gives you the area under the curve to the left of the z-score. Let's call this value A.
A = 0.8944 (estimated from the table)
Step 3: Calculate the probability of the event.
Since you want to find the probability that a randomly chosen male from this group weighs more than 190lb, you need to subtract A from 1.
P(X > 190) = 1 - A = 1 - 0.8944 = 0.1056
So, the probability that a randomly chosen male from this group weighs more than 190lb is approximately 0.1056 or 10.56%.
Alternatively, you can also use a calculator or statistical software to compute this probability directly by entering the mean, standard deviation, and the desired value.
Keep in mind that this calculation assumes that the weights of the adult males follow a normal distribution.