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 solve this problem, we need to follow these steps:
Step 1: Calculate the number of students absent from each class meeting.
Step 2: Calculate the mean number of students absent.
Step 3: Calculate the variance of the number of students absent.
Step 4: Calculate the standard deviation of the number of students absent.
Step 5: Create a probability table and a distribution graph.
Now, let's tackle each step one by one:
Step 1: Calculate the number of students absent from each class meeting.
Given that there is a 12% absentee rate per class meeting, we can calculate the number of absent students as follows:
Number of absent students = 12% of 200
Number of absent students = 0.12 * 200
Number of absent students = 24
Step 2: Calculate the mean number of students absent.
To find the mean, we need to add up the number of absent students from each class meeting and divide it by the total number of class meetings. Since we are considering only a single class meeting, the mean number of students absent will be equal to the number of absent students from that meeting, which is 24.
Step 3: Calculate the variance of the number of students absent.
The variance describes how much the number of absent students varies from the mean. In this case, since we have only one class meeting, the variance will be zero.
Step 4: Calculate the standard deviation of the number of students absent.
The standard deviation measures the average amount of variability in the data. Since the variance is zero due to having only one class meeting, the standard deviation is also zero.
Step 5: Create a probability table and a distribution graph.
To create a probability table and distribution graph, we need to know the range of the number of students absent. As per our calculations, there is only one possible outcome: 24 students absent.
Probability Table:
Number of Students Absent | Probability (assuming one class meeting)
24 | 1
Distribution Graph:
On the x-axis, we plot the number of students absent (with the only possibility being 24), and on the y-axis, we plot the corresponding probability of that event occurring. In this case, we would simply plot a bar at 24 with a height of 1.
That's it! You have now calculated the mean, variance, and standard deviation of the number of students absent from each class meeting, and created a probability table and a distribution graph for the number of students absent.
To find the mean, variance, and standard deviation of the number of students who will be absent from each class, we need to use the formula for a binomial distribution.
a) Calculation:
Let's define the probability of a student being absent as p and the number of trials (class meetings) as n.
1. Mean (μ):
The mean of a binomial distribution is given by the formula: μ = n * p
In this case, p = 0.12 (12% absentee rate) and n = 200 (number of class members). So, substituting the values into the formula, we get:
μ = 200 * 0.12 = 24
Therefore, the mean number of students absent from each class is 24.
2. Variance (σ^2):
The variance of a binomial distribution is given by the formula: σ^2 = n * p * (1 - p)
Substituting the values into the formula, we get:
σ^2 = 200 * 0.12 * (1 - 0.12) = 20.736
Therefore, the variance of the number of students absent from each class is approximately 20.736.
3. Standard deviation (σ):
The standard deviation of a binomial distribution is the square root of the variance: σ = √(σ^2)
Substituting the value of σ^2 into the formula, we get:
σ = √(20.736) ≈ 4.553
Therefore, the standard deviation of the number of students absent from each class is approximately 4.553.
b) Probability table and distribution graph:
To draw the probability table and distribution graph, we need to calculate the probabilities for each possible number of students absent (0 to 200) using binomial distribution formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient.
We can use a statistical software or calculator to compute the probabilities, but let's demonstrate a few examples:
- If k = 0:
P(X = 0) = C(200, 0) * 0.12^0 * (1-0.12)^(200-0) = 0.788 (rounded to 3 decimal places)
- If k = 1:
P(X = 1) = C(200, 1) * 0.12^1 * (1-0.12)^(200-1) = 0.198 (rounded to 3 decimal places)
- If k = 2:
P(X = 2) = C(200, 2) * 0.12^2 * (1-0.12)^(200-2) = 0.046 (rounded to 3 decimal places)
Continue this calculation for each possible number of absent students (0 to 200).
The probability table will have one column for the number of students absent (X), and another column for the corresponding probability (P(X = k)).
The distribution graph can be plotted on a histogram, with the x-axis representing the number of students absent (X) and the y-axis representing the probability (P(X = k)).
Please note that it may not be practical to manually calculate and present the probability table and distribution graph for all 201 possible values of X. However, the concepts described above can be used with appropriate software or tools to generate these representations.