# Assignment 1 (Probability)

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account_balance_wallet \$15

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The instructions and questions is in the attachment below. If you have any questions, please ask. Thanks

MATH 153 T. McKoy Summer 2018 Project 3 – Probability – Distributions - Normal 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. 1. 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? 2. Provide the conditional distribution for the usage category based on the monthly grouping. 3. If one of the BGE bills was selected at random, find a. P(Sep-Dec or Medium Usage) b. P(High Usage and Jan-Apr) MATH 153 T. McKoy Summer 2018 Project 3 – Probability – Distributions - Normal 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 4. Explain why the table above represents a probability distribution. 5. Find P(x < 5). 6. 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? 7. Using the 4 criteria, show why this distribution would represent a binomial experiment. 8. Use the value of p and show the calculation (formula) for finding P(x = 3). 9. Calculate the mean and standard deviation for this binomial distribution using the (short) formulas. Show your work. MATH 153 T. McKoy Summer 2018 Project 3 – Probability – Distributions - Normal NORMAL DISTRIBUTION For the 156 BGE bills collected, the average of the average monthly temperatures is 57.2° and the standard deviation is 15.9°. 10. Use the mean and standard deviation to sketch a normal distribution and use the Empirical Rule to label the areas. 11. Using the distribution just sketched, identify the two average monthly temperatures that would separate usual from unusual temperatures. 12. What would be the Z-score for a temperature of 75°? What percent of the bills had average monthly temperatures higher than this value? 13. What temperature would have a Z-score of -1.25? What percent of the bills had average monthly temperatures below this value? 14. What average monthly temperature values would identify the middle 80% of the bills? 15. Given that the percentage of bills having a high average monthly temperature is 21%, and that the process of selecting bills is considered binomial, if the random variable X represents the number of bills with high average monthly temperatures, show why this distribution can be approximated with the normal distribution. 16. Extra Credit: Knowing that the binomial distribution in #15 can be approximated with the normal distribution, find P(30 < x < 40), the probability of randomly selecting between 30 and 40 bills that have high average monthly temperatures.

Ace_Tutor
School: UIUC

my solution is 100% done, and attached below

MATH 153
T. McKoy

Summer 2018 Project 3 – Probability – Distributions - Normal

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.
1. 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?
The total number of bills categorized as Low Usage is 5 + 33 + 20 = 58 bills, and there is a total of 156
bills. Hence, the probability of the bill being categorized as Low Usage is equal to

P ( Low Usage ) =

58
= 0.3718
156

2. Provide the conditional distribution for the usage category based on the monthly grouping.
The conditional distribution for the usage category can be provided as follow:

MATH 153
T. McKoy

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Anonymous
Awesome! Exactly what I wanted.

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