The weight of 100 females aged 15 to 20 years has a mean of 120 pounds and a variance of 36 pounds. How many of the students has a weight less than 110 pounds?

Z = (score-mean)/SD

SD = √variance

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply by 100.

28.77%

To solve this problem, we need to use the concept of standard deviation.

Standard deviation is the statistical measure of the amount of variation or dispersion in a set of values. It gives an idea of how spread out the values are from the mean.

In this case, we know the variance, which is the square of the standard deviation. So we can first calculate the standard deviation.

The given variance is 36 pounds. To find the standard deviation, we take the square root of the variance:

Standard deviation = √(variance) = √36 = 6 pounds.

Now, we know the mean weight is 120 pounds, and the standard deviation is 6 pounds.

To find how many students have a weight less than 110 pounds, we need to find the z-score of 110 pounds.

The z-score is a measure of how many standard deviations an individual data point is from the mean. It is calculated using the formula:

z = (X - μ) / σ,

where X is the data point, μ is the mean, and σ is the standard deviation.

In this case, X = 110 pounds, μ = 120 pounds, and σ = 6 pounds.

Substituting the values into the formula:

z = (110 - 120) / 6 = -10 / 6 = -1.67

Now that we have the z-score, we need to find the area under the normal distribution curve to the left of this z-score. This area represents the proportion of students with a weight less than 110 pounds.

We can use a standard normal distribution table or a statistical calculator to find this area. For example, using a standard normal distribution table, we can find that the area to the left of -1.67 is approximately 0.0475.

This means that approximately 0.0475 or 4.75% of the students have a weight less than 110 pounds.

Therefore, out of the 100 females aged 15 to 20 years, approximately 4.75 students have a weight less than 110 pounds. Since we can't have a fraction of a student, we can say that about 5 students have a weight less than 110 pounds.