The percent of fat calories that a person consumes each day is normally distributed with a mean of about 37 and a standard deviation of about ten. Suppose that 16 individuals are randomly chosen. Let X = average percent of fat calories.
Find the first quartile for the average percent of fat calories.
http://davidmlane.com/hyperstat/z_table.html
To find the first quartile for the average percent of fat calories, we need to find the z-score corresponding to the first quartile and then convert it back to the original measurement scale.
The first quartile corresponds to the 25th percentile, which means that 25% of the data falls below this value. We can calculate the z-score for the first quartile using the standard normal distribution formula:
z = (X - μ) / σ
where X is the desired value, μ is the mean, and σ is the standard deviation.
In this case, the mean (μ) is 37 and the standard deviation (σ) is 10. The desired value (X) represents the first quartile.
Let's calculate the z-score first:
z = (X - μ) / σ
z = (X - 37) / 10
Next, we need to find the z-score that corresponds to the 25th percentile. This value can be obtained from the standard normal distribution table or by using a statistical software. For the 25th percentile, the z-score is approximately -0.674.
Now, we can solve for X in the z-score formula:
-0.674 = (X - 37) / 10
To isolate X, we can multiply both sides of the equation by 10:
-0.674 * 10 = X - 37
-6.74 = X - 37
To isolate X, we can add 37 to both sides:
X = -6.74 + 37 =
X ≈ 30.26
So, the first quartile for the average percent of fat calories is approximately 30.26.