The height of males is normally distributed with a mean of 69 inches and a standard deviation of 2.5 inches. What is the probability that the height of a randomly selected male is between 66 and 71 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 scores.
To find the probability that the height of a randomly selected male is between 66 and 71 inches, we can use the cumulative distribution function (CDF) of the normal distribution.
The Z-score formula can help us standardize the values and calculate the probabilities. The Z-score of a particular height value, x, can be calculated using the following formula:
Z = (x - mean) / standard deviation
Given that the mean is 69 inches and the standard deviation is 2.5 inches, we can calculate the Z-scores for the lower and upper limits of the height range:
Lower Z-score:
Z_lower = (66 - 69) / 2.5
Upper Z-score:
Z_upper = (71 - 69) / 2.5
Now, we can use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities associated with these Z-scores.
CDF_lower = Φ(Z_lower)
CDF_upper = Φ(Z_upper)
To find the probability that the height is between 66 and 71 inches, we subtract the probability corresponding to the lower Z-score from the probability corresponding to the upper Z-score:
P(66 ≤ height ≤ 71) = CDF_upper - CDF_lower
To get this probability, you can use statistical software or lookup tables for the standard normal distribution to find the probabilities associated with the calculated Z-scores, and then subtract the CDF_lower from CDF_upper.
For example, using a standard normal distribution table, you can look up the values for CDF_lower and CDF_upper. Subtracting the CDF_lower value from the CDF_upper value will give you the desired probability.