Context: It is now time to look at the McKoy family’s BGE data through the lenses of probability and distributions. The list of data is not required for this project. You will look at a two-way table (contingency table) based on months and average daily usage, as well as a list of random variable outcomes and the probability of their occurrences. Finally, the binomial and the normal distributions are applied to answer questions about the probability of specific outcomes. As usual, provide complete and clear responses to the questions below.
PROBABILITY
Low Usage | Medium Usage | High Usage | Total | |
Jan – Apr | 5 | 20 | 27 | 52 |
May – Aug | 33 | 19 | 0 | 52 |
Sep – Dec | 20 | 26 | 6 | 52 |
Total | 58 | 65 | 33 | 156 |
The above contingency table summarizes the average daily usages on all of the McKoy family’s BGE bills from January 2005 to December 2017. The years were divided into 3 equal parts consisting of four months and this is represented by the first column. Each row shows the number of months in the grouping that was classified by a specific usage category (low, medium, or high). Of course, the entire table shows the interaction between these two variables. Use this table to respond to the following questions.
- If one of the BGE bills was selected at random, based on the marginal distributions, what is the probability of the bill being categorized as Low Usage?
- Provide the conditional distribution for the usage category based on the monthly grouping.
- If one of the BGE bills was selected at random, find
- P(Sep-Dec or Medium Usage)
- P(High Usage and Jan-Apr)
PROBABILITY DISTRIBUTIONS
In the table below the random variable X represents the number of BGE bills that had Medium Usage values when 8 bills were selected at random. Of course, P(x) represents the probability of each possible outcome. Use this table to respond to the questions that follow.
x | P(x) |
0 | 0.0134 |
1 | 0.0766 |
2 | 0.1915 |
3 | 0.2736 |
4 | 0.2443 |
5 | 0.1396 |
6 | 0.0499 |
7 | 0.0102 |
8 | 0.0009 |
- Explain why the table above represents a probability distribution.
- Find P(x < 5).
- Based on this distribution of the outcomes when 8 bills are selected at random, what is the expected number that would fall into the Medium Usage category?
- Using the 4 criteria, show why this distribution would represent a binomial experiment.
- Use the value of p and show the calculation (formula) for finding P(x = 3).
- Calculate the mean and standard deviation for this binomial distribution using the (short) formulas. Show your work.