80. Florida State University has 14 statistics classes scheduled for its Summer 2013 term. One class has space available for 30 students, eight classes have space for 60 students, one class has space for 70 students, and four classes have space for 100 students.
a. What is the average class size assuming each class is filled to capacity?
b. Space is available for 980 students. Suppose that each class is filled to capacity and select a statistics student at random. Let the random variable X equal the size of the student’s class. Define the PDF for X.
c. Find the mean of X.
d. Find the standard deviation of X.
mean = ∑x/n = 980/14 = ?
Subtract each of the scores (# in each class) from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
a. Well, let's do some math gymnastics. The total number of students that can be accommodated in all the classes is:
(30 * 1) + (60 * 8) + (70 * 1) + (100 * 4)
= 30 + 480 + 70 + 400
= 980
And since there are a total of 14 classes, we can find the average class size by dividing the total number of students by the number of classes:
Average class size = 980 / 14 = 70
So, the average class size assuming each class is filled to capacity is 70 students.
b. To define the probability density function (PDF) for the random variable X, we need to consider the class sizes and their respective probabilities.
There is one class with space for 30 students, eight classes with space for 60 students, one class with space for 70 students, and four classes with space for 100 students.
The probabilities associated with each class size are:
P(X = 30) = 1/14
P(X = 60) = 8/14
P(X = 70) = 1/14
P(X = 100) = 4/14
So, the PDF for X would be:
f(x) =
1/14, for x = 30
8/14, for x = 60
1/14, for x = 70
4/14, for x = 100
c. To find the mean of X, we can multiply each class size by its respective probability and sum them up:
Mean(X) = (30 * 1/14) + (60 * 8/14) + (70 * 1/14) + (100 * 4/14)
= 30/14 + 480/14 + 70/14 + 400/14
= 980/14
= 70
Therefore, the mean of X is 70.
d. To find the standard deviation of X, we need to use the formula:
Standard deviation = sqrt(variance)
The variance can be calculated by:
Variance(X) = (30 - 70)^2 * 1/14 + (60 - 70)^2 * 8/14 + (70 - 70)^2 * 1/14 + (100 - 70)^2 * 4/14
= (-40)^2 * 1/14 + (-10)^2 * 8/14 + 0^2 * 1/14 + 30^2 * 4/14
= 1600/14 + 800/14 + 0/14 + 900/14
= 3200/14 + 800/14 + 900/14
= 4900/14
= 350
So, the variance of X is 350.
And the standard deviation of X is:
Standard deviation = sqrt(350)
Therefore, the standard deviation of X is approximately 18.71.
a. To find the average class size, we need to calculate the weighted average.
Class 1: 30 students (1 class)
Class 2-9: 60 students each (8 classes)
Class 10: 70 students (1 class)
Class 11-14: 100 students each (4 classes)
Total number of students = (30*1) + (60*8) + (70*1) + (100*4) = 770
Total number of classes = 14
Average class size = Total number of students / Total number of classes = 770 / 14 = 55
Therefore, the average class size is 55 students.
b. To define the probability density function (PDF) for the random variable X (size of student's class), we need to consider the number of students in each class and their probabilities.
Class 1: 30 students (1 class) - Probability = 1/14
Class 2-9: 60 students each (8 classes) - Probability = 8/14
Class 10: 70 students (1 class) - Probability = 1/14
Class 11-14: 100 students each (4 classes) - Probability = 4/14
PDF for X:
X = 30 with probability 1/14
X = 60 with probability 8/14
X = 70 with probability 1/14
X = 100 with probability 4/14
c. To find the mean of X, we need to multiply each class size by its corresponding probability and sum them up.
Mean of X = (30 * 1/14) + (60 * 8/14) + (70 * 1/14) + (100 * 4/14)
= 30/14 + 480/14 + 70/14 + 400/14
= 980/14
= 70
Therefore, the mean of X is 70.
d. To find the standard deviation of X, we need to calculate the variance first. The variance is the average of the squared deviations from the mean.
Variance of X = [(30-70)^2 * 1/14] + [(60-70)^2 * 8/14] + [(70-70)^2 * 1/14] + [(100-70)^2 * 4/14]
= 40^2/14 + (-10)^2 * 8/14 + 0^2 * 1/14 + 30^2 * 4/14
= 1600/14 + 100*8/14 + 0 + 900*4/14
= 114.2857 + 57.1429 + 257.1429
= 428.5715
Standard deviation of X = square root of variance of X = square root of 428.5715
≈ 20.71
Therefore, the standard deviation of X is approximately 20.71.
a. To find the average class size, we need to calculate the total number of students and divide it by the total number of classes.
The number of students in each class is as follows:
- 1 class with space for 30 students = 30 students
- 8 classes with space for 60 students = 8 * 60 = 480 students
- 1 class with space for 70 students = 70 students
- 4 classes with space for 100 students = 4 * 100 = 400 students
Total number of students = 30 + 480 + 70 + 400 = 980 students
Total number of classes = 14 classes
Average class size = Total number of students / Total number of classes
= 980 / 14
= 70 students
Therefore, the average class size assuming each class is filled to capacity is 70 students.
b. To define the PDF (Probability Density Function) for the random variable X, we need to calculate the probability of each class size occurring.
The class sizes and the number of classes for each size are as follows:
- Class size 30: 1 class
- Class size 60: 8 classes
- Class size 70: 1 class
- Class size 100: 4 classes
Total number of classes = 14 classes
PDF for X:
P(X = 30) = 1/14
P(X = 60) = 8/14
P(X = 70) = 1/14
P(X = 100) = 4/14
Therefore, the PDF for X is:
P(X = 30) = 1/14
P(X = 60) = 8/14
P(X = 70) = 1/14
P(X = 100) = 4/14
c. The mean of X can be calculated by multiplying each class size by its corresponding probability and summing them up.
Mean of X = (30 * 1/14) + (60 * 8/14) + (70 * 1/14) + (100 * 4/14)
= 30/14 + 480/14 + 70/14 + 400/14
= 980/14
= 70 students
Therefore, the mean of X is 70 students.
d. The standard deviation of X measures the dispersion of the class sizes. It can be calculated using the formula:
Standard deviation of X = sqrt([(class size - mean of X)^2 * probability] summed for all class sizes)
Using the values from the previous calculations, the standard deviation of X is:
Standard deviation of X = sqrt([((30 - 70)^2 * 1/14) + ((60 - 70)^2 * 8/14) + ((70 - 70)^2 * 1/14) + ((100 - 70)^2 * 4/14)])
= sqrt([40^2 * 1/14 + 10^2 * 8/14 + 0^2 * 1/14 + 30^2 * 4/14])
= sqrt([1600/14 + 800/14 + 0 + 3600/14])
= sqrt([41600/14])
= sqrt(2971.43)
≈ 54.49
Therefore, the standard deviation of X is approximately 54.49.