To find the proportion of male college students whose height is greater than 69 inches, we need to use the standard normal distribution.
(b) Calculating the proportion:
1. Convert the given value of 69 inches to a z-score. The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the observed value, μ is the mean, and σ is the standard deviation.
In this case, x = 69 inches, μ = 69 inches, and σ = 4.6 inches.
Plugging in the values, we get:
z = (69 - 69) / 4.6 = 0
2. Look up the z-score in the standard normal distribution table (also known as the Z-table) to find the corresponding proportion.
The proportion corresponding to a z-score of 0 is 0.5000.
However, we need to find the proportion whose height is greater than 69 inches, not equal to 69 inches. Since the standard normal distribution is symmetrical, we can subtract 0.5000 from 1 to get the desired proportion:
Proportion = 1 - 0.5000 = 0.5000
So, the proportion of male college students whose height is greater than 69 inches is 0.5000.
(e) To find P(x > 70), we repeat the same steps as above.
1. Convert the value of 70 inches to a z-score:
z = (70 - 69) / 4.6 = 0.2174 (rounded to four decimal places)
2. Look up the z-score in the standard normal distribution table to find the corresponding proportion.
From the z-table, we find that the proportion corresponding to a z-score of 0.2174 is 0.5868.
However, we need to find P(x > 70), which means finding the proportion greater than 70 inches. Since the standard normal distribution is symmetrical, we subtract the obtained proportion from 1 to get the desired proportion:
Proportion = 1 - 0.5868 = 0.4132
So, P(x > 70) is approximately 0.4132.
(f) To find P(x < 67), we again repeat the same steps.
1. Convert the value of 67 inches to a z-score:
z = (67 - 69) / 4.6 = -0.4348 (rounded to four decimal places)
2. Look up the z-score in the standard normal distribution table to find the corresponding proportion.
From the z-table, we find that the proportion corresponding to a z-score of -0.4348 is 0.3325.
So, P(x < 67) is approximately 0.3325.