A mathematics class has 200 members. If there is 12% absentee rate per class meeting, find:
a) mean, variance and standard deviation number of students who will be absent from each class.
b) Draw the probability table and distribution graph of the number of the students absent.
To find the mean, variance, and standard deviation of the number of students who will be absent from each class, we can use the following formulas:
a) Mean:
The mean is the average number of students who will be absent from each class. We can find it by multiplying the class size (200) by the absentee rate (12% or 0.12):
Mean = Class size * Absentee rate
Mean = 200 * 0.12
Mean = 24
b) Variance:
The variance is a measure of how spread out the numbers are. It can be calculated by multiplying the square of the standard deviation by the mean:
Variance = (Absentee rate * Class size) * (1 - Absentee rate)
Variance = (0.12 * 200) * (1 - 0.12)
Variance = 24 * 0.88
Variance = 21.12
c) Standard deviation:
The standard deviation is the square root of the variance:
Standard deviation = √Variance
Standard deviation = √21.12
Standard deviation ≈ 4.6
Now, let's move on to drawing the probability table and distribution graph for the number of students absent:
b) Probability table:
Since the absentee rate is given as a percentage, we can assume that it follows a binomial distribution. The number of students absent can range from 0 to 200, and the probability of each number is given by:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of k students being absent,
- C(n,k) is the number of combinations,
- p is the absentee rate,
- n is the class size (200 in this case), and
- k is the number of students absent.
Using this formula, we can calculate the probabilities for each value of k (from 0 to 200) and construct the probability table.
To create the probability distribution graph, we can plot the number of students absent on the x-axis and the corresponding probabilities on the y-axis.
Unfortunately, due to the lengthy calculations required and the limitations of this text-based format, I am unable to provide the complete probability table and distribution graph here. However, you can use the given information and the formulas provided to perform the calculations and plot the graph yourself.
To find the mean, variance, and standard deviation of the number of students who will be absent from each class, we can follow these steps:
a) Mean:
The mean, also known as the expected value, is the average number of students who will be absent.
To find the mean:
Step 1: Calculate the number of students that will be absent for each class meeting.
12% of 200 members = 0.12 * 200 = 24 students
Step 2: Divide the total number of absent students by the number of class meetings.
Mean = Total absent students / Number of class meetings
The mean for the number of students who will be absent from each class is 24.
b) Variance:
The variance measures how spread-out the data is. In this case, it tells us how much the number of absent students varies from the mean.
To find the variance:
Step 1: Calculate the difference between the number of absent students and the mean for each class meeting.
Difference = Number of absent students - Mean
Step 2: Square each difference.
Step 3: Calculate the average of the squared differences.
Variance = Average of squared differences
The variance for the number of students who will be absent from each class is (0^2 + 0^2 + ... + 0^2) / Number of class meetings = 0.
c) Standard Deviation:
The standard deviation is a measure of the dispersion or spread of the data. It is the square root of the variance.
To find the standard deviation:
Standard Deviation = Square root of the Variance
The standard deviation for the number of students who will be absent from each class is √0 = 0.
b) Probability Table and Distribution Graph:
To draw the probability table and distribution graph of the number of students absent, we need to calculate the probability of each possible number of absent students.
In this case, we have only one possibility - 24 absent students - as the mean is 24.
Probability Table:
Number of Absent Students | Probability
------------------------- | ------------
24 | 1
Distribution Graph:
To create a distribution graph, plot the number of absent students on the x-axis and the corresponding probability on the y-axis. Since we only have one possible number - 24 - the graph will consist of a bar at x=24 with a height of 1.
Note: In this particular scenario, due to the fixed percentage and class size, there is no variation in the number of absent students from class to class, resulting in a variance and standard deviation of 0.