9. Assume the weight, W, of a randomly selected adult has a normal distribution with a mean

of 175 pounds and a standard deviation of 10 pounds.. Find the probability, P(W < 190), that
a randomly selected adult weighs less than 190 pounds.

http://davidmlane.com/hyperstat/z_table.html

Another method.

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 selected adult weighs less than 190 pounds, we need to calculate the cumulative probability up to 190 pounds using the given mean and standard deviation.

The standard normal distribution (also known as the z-distribution) can be used to find these probabilities. We need to standardize the value of 190 pounds using the formula:

z = (x - μ) / σ

Where:
- z is the standardized value,
- x is the observed value,
- μ is the mean of the distribution, and
- σ is the standard deviation of the distribution.

Let's calculate z using the given values:

z = (190 - 175) / 10
= 15 / 10
= 1.5

Now, we need to find the cumulative probability up to z = 1.5. We can refer to the standard normal distribution table or use a statistical software to find this value. Assuming a standard normal distribution table is being used, the cumulative probability up to z = 1.5 is approximately 0.9332.

Therefore, the probability that a randomly selected adult weighs less than 190 pounds, P(W < 190), is approximately 0.9332 or 93.32%.

To find the probability that a randomly selected adult weighs less than 190 pounds, we need to calculate the area under the normal distribution curve up to 190 pounds.

To do this, we need to standardize the value of 190 pounds using the Z-score formula:

Z = (X - μ) / σ

Where:
X is the value we want to standardize (190 pounds in this case)
μ is the mean of the distribution (175 pounds)
σ is the standard deviation of the distribution (10 pounds)

Plugging in the values, we get:

Z = (190 - 175) / 10
Z = 15 / 10
Z = 1.5

Now, we need to find the corresponding area in the standard normal distribution table for a Z-score of 1.5.

Looking up the value of 1.5 in the Z-table, we find that the area under the curve up to 1.5 is approximately 0.9332.

Therefore, the probability P(W < 190) is approximately 0.9332 or 93.32%.