7. Most powerful business women in America. Consider that the data set is a normal distribution with a mean of 50 years and a standard deviation of 5 years. A powerful woman is randomly selected from the data, and her age is observed.
Please calculate the probability of the woman having less than 55 years. Also calculate the probability of the woman having between 40 and 60 years, and the probability of the woman being more than 55 years old.
use this webpage instead of tables of normal distribution values
http://davidmlane.com/normal.html
enter:
mean: 50
SD: 5
click for the appropriate question
e.g. prob of woman being between 40 and 60,
click on "between" bubble and enter 40 and 60 to get
.9545 or 95.45%
To calculate the probabilities in this scenario, we can use the properties of a normal distribution. Specifically, we will use the concept of standardization, which involves converting the observed values into z-scores. The z-score represents the number of standard deviations a value is away from the mean.
1. Probability of the woman having less than 55 years:
We can calculate this probability by finding the area under the normal distribution curve to the left of 55 years.
First, we calculate the z-score for 55 years:
z = (x - mean) / standard deviation
z = (55 - 50) / 5
z = 1
Now, we need to find the corresponding area under the standard normal distribution curve for a z-score of 1. We can use a z-table or a statistical software to find this probability. Looking up the z-score of 1 in a standard normal distribution table, we find that the area to the left of 1 is approximately 0.8413.
Therefore, the probability of the woman having less than 55 years is approximately 0.8413.
2. Probability of the woman having between 40 and 60 years:
To find this probability, we need to calculate the area under the normal distribution curve between 40 and 60 years.
First, we calculate the z-scores for 40 and 60 years:
For 40 years:
z1 = (40 - 50) / 5
z1 = -2
For 60 years:
z2 = (60 - 50) / 5
z2 = 2
Next, we need to find the corresponding areas under the standard normal distribution curve for z1 and z2. Using the z-table, the area to the left of -2 is approximately 0.0228 and the area to the left of 2 is approximately 0.9772.
To find the area between z1 and z2, we subtract the area to the left of z1 from the area to the left of z2:
P(40 < x < 60) = P(z1 < z < z2) = 0.9772 - 0.0228
P(40 < x < 60) = 0.9544
Therefore, the probability of the woman having between 40 and 60 years is approximately 0.9544.
3. Probability of the woman being more than 55 years old:
To find this probability, we need to calculate the area under the normal distribution curve to the right of 55 years.
Since we have already calculated the z-score for 55 years earlier (which is 1), we can use the complementary probability (1 - probability of less than 55) to find the probability of being more than 55 years.
P(x > 55) = 1 - P(x < 55)
P(x > 55) = 1 - 0.8413
P(x > 55) = 0.1587
Therefore, the probability of the woman being more than 55 years old is approximately 0.1587.