Florida State University has 14 statistics classes scheduled...

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 $X.$

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 $X.$

Key Concepts

Expected Value

The expected value of a random variable is the probability-weighted average of all possible values. It represents the long-run average outcome if the random process were to be repeated many times and is computed by summing the product of each outcome with its probability.

Probability Distribution Function (PDF)

A probability distribution function for a discrete random variable assigns a probability to each possible outcome. It provides a complete description of the likelihood of the variable taking on each of its possible values and is essential for further statistical analysis.

Variance and Standard Deviation

Variance measures the spread of a set of values by calculating the average squared deviation from the mean, while the standard deviation is the square root of the variance. Together, these measures indicate the extent to which individual outcomes tend to deviate from the mean, thus quantifying the dispersion in the data.

Weighted Average

A weighted average is an average where each value to be averaged is multiplied by a weight that reflects its relative importance. It is particularly useful in situations where different groups or classes contribute unequally to the overall sum, such as when each class has a different capacity.

Arithmetic Mean

The arithmetic mean is a measure of central tendency calculated by summing a set of numbers and dividing by the count of those numbers. In problems like this, it represents the overall average by considering total capacity and the number of classes or groups, providing a simple measure of central location.

Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In this context, it represents the size of the class that a randomly selected student belongs to, taking on discrete values with associated probabilities.

Transcript

0:00 80.

00:02 Florida state university has 14 statistics classes scheduled for its summer 2013 term.

00:08 One class has space available for 30 students.

00:12 Eight classes have space for 60 students.

00:15 One class has space for 70 students.

00:17 And four classes have space for 100 students.

00:21 Question a.

00:22 What is the average class size assuming each class is filled to capacity? because in this problem we're interested in the class size, we're going to let our random variable x be represented with 30, 60, 70, 100 to represent the students in each class.

00:40 Because we have 14 classes, we can represent the probabilities of each of those students ' class sizes to be 1 out of 14 for 30, 8 out of 14 for 60, 1 out of 14 for 70, and 4 out of 14 for 100.

00:59 So the question is saying, what is the average class size? now, average, or mu, is found by adding together each x value multiplied with its probability.

01:13 So we have our 30 multiplied by its 1 out of 14, 60 multiplied by 8 out of 14, 70 multiplied both 1 out of 14, and 100 times 4 out of 14.

01:27 We can add together these four products, and we get an average class size.

01:35 Of 70.

01:41 B.

01:43 Space is available for 980 students.

01:46 Suppose that each class is filled to capacity and select a statistic student at random.

01:51 Let the random variable x equal the size of the student's class define the pdf for x.

01:59 So because we've already found our x values and our probabilities of those x values, we're just going to put those in the table to create our probability distribution.

02:11 We know that our x values were 30, 60, 70, and 100.

02:22 Their probability of 30 was 1 out of 14...

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