W#3 NOTE Probability Distribution of Random Variables

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Fairleigh Dickinson University *

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2023

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Statistics

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Apr 3, 2024

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W#3 NOTE Probability Distribution of Random Variables Instructions The management of the Café Corner chain runs a rewards or loyalty program for its most frequent customers who visit their coffee shops weekly. During a national crisis, the chief operations officer wondered if patron traffic among this very loyal group of customers was diminished. She gathered information from computerized cash register records, and devised the following probability distribution table from relative frequency data about how many days per week these customers stopped by and made purchases. Let x represents the number of days per week these patrons visited the coffee shop and bought products: x 0 1 2 3 4 5 6 7 P % B% [6% B% [10% [12% 0% [222 She forgot to enter the last probability. Complete this probability distribution table and answer the following questions: [HINT: Express all answer as percentages using the percent symbol. Ex: 1% or 99%.] Question 1 0.5 out of 0.5 points The algebraic symbols, such as x, that can describe a probability event are considered a random variable representing that event. This can be written as P(x). Agree or disagree? Selected Answer: Agree Question 2 0.5 out of 0.5 points For this particular context about the coffee shop chain and its most frequent customers, the number of visits per week ending in a purchase could be considered the random variable, say Xx. The values this random variable could assume in this context are 0,1,2,3,4,5,6, and 7, since these most frequent customers could make any number of days, from 0 to 7. Agree or disagree? Selected Answer: Agree Question 3 0.5 out of 0.5 points For this particular context about the coffee shop chain and its most frequent customers making purchases over the course of a week, the random variable x could only take on eight counting number answers, from 0 to 7. It could not, for example, take on the value 3.5 or 9 or -2. Those values don't make sense in this context. Agree or disagree? Selected Answer: Agree Question 4 0.5 out of 0.5 points Because this probability distribution assumes only selected values, which are often counting number values as in this case, and not a spectrum of values, the random variable should be considered discrete rather than continuous. Agree or disagree? Selected Answer: Agree Question 5 0.5 out of 0.5 points Reading the probability table summarizing this distribution for this random variable, x, the top row indicates the value of the random variable, and the bottom row the corresponding probability, P(x). kK o [ R B [ [ [ [ Px) 2% [3% [5% [8% [10% [12% [20% [22? Agree or disagree? Selected Answer: Agree Question 6 0.5 out of 0.5 points According to the probability table for this random variable, the chance of a randomly selected frequent customer making purchases five days in a week during the crisis is 10%. X 0 1 2 4 5 6 7 |3 Px) 2% [3% [6% |8% 10% [12% |20% |??? Agree or disagree? Selected Answer: Disagree Question 7 0.5 out of 0.5 points According to the probability table for this random variable, and writing the question symbolically, P(x=2) = 5%. X 0 1 2 3 4 5 6 7 P(x) 2% 3% [5% 8% [10% |12% [20% |??? Agree or disagree? Selected Answer: Agree

Question 8 1 out of 1 points Express the sum of all eight probabilities associated with this random variable as a whole percentage with percent symbol. [HINT: The last one is missing, but include the value you expect it must be in your calculation.] x o 1 2 [3 4 5 6 7 Px) % [3% [5% [8% [10% [12% [20% [??7? Selected Answer: 100% Question 9 2 out of 2 points What is the chance a most frequent customer made purchases on seven days during the crisis? x 0 1 2 3 4 5 6 7 PX) 2% [3% [5% [8% [10% [12% |20% 2?7 Selected Answer: 40% Question 10 2 out of 2 points Determine P(x # 2) : x o [ 2 B [ [ b PX) [2% [3% [6% [8% [10% [12% [20% [?7? Selected Answer: 95% Question 11 2 out of 2 points What is the chance a most frequent customer made purchases on either five or six days during the crisis? X 0 1 2 3 4 5 6 7 PX) 2% 3% [5% [8% [10% [12% [20% [??? Selected Answer: 32%

Question 12 2 out of 2 points What is the chance a most frequent customer made purchases on neither five nor six days during the crisis? x o [ 2 B & [ 8 |7 Px) 2% [3% [6% [8% [10% [12% [20% [2?7? Selected Answer: 68% Question 13 2 out of 2 points Determine P(x > 4) . x o 1 [2 [3 4 5 6 7 Px) [2% [% [5% [8% [10% [12% [20% [?7? Selected Answer: 82% Question 14 2 out of 2 points What is the chance a most frequent customer made purchases on no more than four days during the crisis? x o 1 2 [3 4 5 /6 7 Px) [2% [3% [5% [8% [10% [12% [20% [??? Selected Answer: 28% Question 15 2 out of 2 points What is the chance a most frequent customer made purchases on fewer than six days during the crisis? x o 1 2 [3 4 5 6 7 Px) % [3% [5% [8% [10% [12% [20% [??2? Selected Answer: 40% Question 16 2 out of 2 points Determine P( 1<x< 7) . x o 1 2 [3 4 5 6 7 Px) [2% [3% [5% [8% [10% [12% [20% [772 Selected Answer: 58% Question 17 2.5 out of 2.5 points On average, how many days did these patrons stop by to make purchases during the week of the crisis? [HINTS: What is the expectation or expected value for this random variable? Express the answer rounded to the nearest whole number of days. Enter only as a counting number: report approximately 8.49 days only as 8, or approximately 9.50 days only as 10.] Selected Answer: 5 Question 18 2 out of 2 points What shape best describe the probability distribution? x 0 1 2 3 4 5 6 7 PX) 2% [3% [5% [8% [10% [12% [20% |??? Selected Answer: b. response (b): j-shaped

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W#3 NOTE Probability Distribution of Random Variables

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