Calculating Mean and Variance for Graduation Ceremony Guest Tickets

In summary, the conversation discusses the mean and variance of the number of tickets required by graduating students for a ceremony, with 70% wanting 2 guest tickets, 20% wanting 1 guest ticket, and 10% not wanting any guest tickets. The mean is calculated to be 1.4 and the variance to be 0.1076. The conversation also mentions using the Central Limit Theorem to solve for the probability that there will not be enough tickets to satisfy demand if 500 guest tickets are available for a ceremony with 300 graduating students.
  • #1
vikkisut88
34
0

Homework Statement


Suppose that, on average, 70% of graduating students want 2 guest tickets for a graduation ceremony, 20% want 1 guest ticket and the remaining10% don't want any guest tickets.

(a) Let X be the number of tickets required by a randomly chosen student. Find the mean and variance of X.

(b) If 500 guest tickets are available for a ceremony at which 300 students are graduating, what is the probability that there will not be enough tickets to satisfy demand?


Homework Equations


Variance = E(X2) - E2(X)


The Attempt at a Solution


I've got the mean to be ((0.7*2) + (0.2*1) + (0.1*0))/3 = 0.53and the variance to be ((0.72*2) + (0.22*1) + (0.12*0))/3 - 0.532 = 0.7391 but this seems wrong because the variance is then bigger than the mean?!
 
Physics news on Phys.org
  • #2
Okay I have since realized that for part a) I think i was doing it wrong so now for the mean I have:

((0.7*2) + (0.2*1) + (0.1*0))/3 = 0.53

But for the Variance I have:

((0.7 - 0.53)2+ (0.2 - 0.53)2 + (0.1 - 0.53)2)/3 = 0.1076 which makes far more sense!

Now I'm thinking of using the Central Limit Theorem to solve b)??!
 
  • #3
vikkisut88 said:
Okay I have since realized that for part a) I think i was doing it wrong so now for the mean I have:

((0.7*2) + (0.2*1) + (0.1*0))/3 = 0.53
This is still incorrect. Suppose every student wanted one ticket. Would the average number be ((0.7*1) + (0.2*1) + (0.1*1))/3 = 1/3? Of course not. The ".7", ".2", and ".1" are already "averages". Don't divide by 3: 0.6*2+ 0.2*1+ 0.1*0= 1.4.

But for the Variance I have:

((0.7 - 0.53)2+ (0.2 - 0.53)2 + (0.1 - 0.53)2)/3 = 0.1076 which makes far more sense!
There is no reason in the world why the variance can't be larger than the mean.

Now I'm thinking of using the Central Limit Theorem to solve b)??!
 
  • #4
Right okay, so now for the variance do I use the same formula but use 1.6 (I presume it was just a typo and it should have been 0.7 *2) instead of 0.53?
Do I then use the Central Limit Theorem for part b?
 

1. What is the definition of mean and variance?

Mean is a measure of central tendency that represents the average value of a set of data. It is calculated by adding all the numbers in the data set and dividing by the total number of values. Variance, on the other hand, measures the spread of the data around the mean. It is calculated by finding the difference between each data point and the mean, squaring those differences, and then finding the average of those squared differences.

2. How do you find the mean and variance of a data set?

To find the mean of a data set, you add up all the numbers in the set and divide by the total number of values. To find the variance, you first find the mean, then subtract the mean from each data point, square those differences, and find the average of those squared differences. This average is the variance. Alternatively, you can use the formula for variance, which is Σ(x-μ)²/n, where Σ represents the sum, x is each data point, μ is the mean, and n is the total number of values.

3. What is the importance of calculating mean and variance?

Calculating mean and variance allows you to summarize and analyze data. Mean gives you an idea of the central value of the data, while variance tells you how spread out the data is. This information is crucial in making decisions and drawing conclusions from the data. For example, if the mean and variance of a data set are high, it indicates that the data is more spread out and may not be representative of the entire population.

4. Can the mean and variance be affected by outliers?

Yes, the mean and variance can be affected by outliers. Outliers are data points that are significantly different from the rest of the data set. Since mean and variance are calculated using all the data points, outliers can pull the values in one direction, making them less representative of the entire data set. It is important to identify and handle outliers properly to get accurate measures of mean and variance.

5. How can mean and variance be used in hypothesis testing?

In hypothesis testing, the mean and variance are used to determine whether there is a significant difference between two or more groups. The mean is used to compare the central values of the groups, while the variance is used to compare the spread of the data. By calculating the mean and variance of each group, researchers can determine if the differences between the groups are due to chance or if there is a significant difference that can be attributed to the independent variable being tested.

Similar threads

Find mean and variance of a random vector
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Small sample test for the difference between two means
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Mean and variance of a probability density
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Calculating mean and variance of probability
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Calculating Variance in Normal Distribution with Given Percentile
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
How do I calculate variance for volume, 𝑉 (i.e, ⟨Δ𝑉2⟩=⟨𝑉2⟩−⟨𝑉⟩2)?
    • Advanced Physics Homework Help
Replies
2
Views
1K
Optimizing Point Estimates: Bias and Variance Analysis for Mean Estimation
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Mean, variance and correlation function of Langevin equation
    • Advanced Physics Homework Help
Replies
2
Views
2K
Sampling Distribution of Mean for Discrete Uniform Distribution with Replacement
    • Calculus and Beyond Homework Help
Replies
12
Views
2K
I Calculate the mean, variance, stdev, and CV with 2 methods: check answers
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K

Hot Threads

Recent Insights

Back
Top