ACT 610 Mathematical Statistics - Homework 1: Probability and Distribution
ACT610 Homework #1
1. Identify the following experimental designs. Are they (i) categorical, (ii) two-sample, (iii) k-sample, (iv) paired, or (v) randomized block?
a. 30 monkeys, 15 of them received an experimental vaccine, and the other 15 received placebo, before all of them were exposed to a particular virus. Four weeks later, in the experiment group, 10 monkeys show marked improvement while 5 do not; in the control group, 7 monkeys show marked improvement while 8 do not.
b. 30 monkeys, 15 of them received an experimental vaccine, and the other 15 received placebo, before all of them were exposed to a particular virus. Four weeks later, they are tested for the amount of antibody. Below are the results:
Monkey ID# | Experiment group | Monkey ID# | Control group |
1 | 0.7 | 16 | 2.1 |
2 | 0.8 | 17 | 2.8 |
3 | 1.2 | 18 | 0.9 |
4 | 2.5 | 19 | 0.6 |
5 | 2.8 | 20 | 0.8 |
6 | 1.1 | 21 | 1.7 |
7 | 0.5 | 22 | 0.3 |
8 | 3.0 | 23 | 0.5 |
9 | 2.2 | 24 | 0.1 |
10 | 2.9 | 25 | 2.0 |
11 | 0.9 | 26 | 2.4 |
12 | 0.6 | 27 | 1.2 |
13 | 0.8 | 28 | 1.3 |
14 | 1.9 | 29 | 1.5 |
15 | 2.8 | 30 | 0.6 |
c. 15 monkeys from 5 families, receive either a placebo, or one dose of vaccine, or two doses of vaccine. The experiment is conducted such that each family will have exactly one member receive each of the three possible treatments. Four weeks later, all monkeys are tested for the amount of antibody. Below are the results:
Control group | One dose | Two doses | |
Family #1 | 0.3 | 2.3 | 0.5 |
Family #2 | 2.2 | 2.4 | 1.1 |
Family #3 | 1.9 | 3.5 | 2.0 |
Family #4 | 0.9 | 2.6 | 2.4 |
Family #5 | 0.6 | 1.7 | 1.2 |
2. An “eyes-only” diplomatic message is to be transmitted as a binary code of 0’s and 1’s. Past experience with the equipment being used suggests that if a 0 is sent, it will be (correctly) received as a “0” 90% of the time (and mistakenly decoded as a “1” 10% of the time). If a “1” is sent, it will be received as a “1” 95% of the time (and as a “0” 5% of the time). The text being sent is 70% 1’s and 30% 0’s. Suppose the next signal sent is received as a “1”. What is the probability that it was sent as a “0”?
3. Amanda is trying to add a little zing to her cabaret act by telling four jokes at the beginning of each show. Her current engagement is booked to run 120 days. If she gives one performance a night and never wants to repeat the same set of jokes on any two nights (i.e., at least one of the four jokes is different), what is the minimum number of jokes she needs in her repertoire?
4. Please submit an Excel file for this question. Create five separate sheets, one for each of the following distributions. For each probability distribution: clearly indicate the parameter values (you can choose whatever values you’d like, e.g., n=50 and p=0.95 for binomial distribution); be aware of the support of the distribution (e.g., 0, 1, 2, …n for binomial); calculate values for probabilities or probability densities; and create a distribution graph (insert column charts). When the support is infinite, create a portion of the theoretical distribution graph that is representative of the entire distribution. When the distribution is continuous, choose appropriate intervals to approximate the distribution. When calculating values for probabilities or probability densities, please type in the probability density/mass function instead of using the built-in excel functions (such as norm.dist).
a. Binomial distribution
b. Poisson distribution
c. Negative Binomial distribution
d. Approximate normal distribution
e. Approximate Gamma distribution (choose r to be an integer)
5. Add two more sheets to your Excel file above. Compare the probability distribution of the random variable that is the mean of 30 dice rolls and the distribution of the random variable that is the mean of 100 dice rolls. What do you find?
