The heights of young women are approximately normal with a mean 64.5 in and a standard deviation of 2.5 in. What is the probability that a woman chosen at random is taller than 63.6 inches?
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 determine the probability that a woman chosen at random is taller than 63.6 inches, we need to use the normal distribution. We can use the Z-score formula to calculate the probability.
The Z-score formula is:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we want to find the probability for (63.6 in this case)
μ is the mean (64.5 in this case)
σ is the standard deviation (2.5 in this case)
Plugging the values into the formula, we get:
Z = (63.6 - 64.5) / 2.5
Calculating this, we get:
Z = -0.36
Now, we need to find the cumulative probability corresponding to this Z-score using a Z-table or calculator. The cumulative probability gives us the proportion of the area under the normal distribution curve up to the given Z-score.
Looking up the Z-score of -0.36 in a standard normal distribution table or using a calculator, we find that the cumulative probability is approximately 0.3594.
However, we are interested in the probability of being taller than 63.6 inches, which is the complement of being shorter than or equal to 63.6 inches. Thus, we subtract the cumulative probability from 1 to get the final probability:
P(X > 63.6) = 1 - 0.3594 = 0.6406
So, the probability that a woman chosen at random is taller than 63.6 inches is approximately 0.6406 or 64.06%.