Question

In: Statistics and Probability

In Unit 2, you have learned about three different types of distributions: Normal, binomial, and Poisson....

In Unit 2, you have learned about three different types of distributions: Normal, binomial, and Poisson. You can take data that you collect and plot it out onto graphs to see a visual representation of the data. By simply looking at data on a graph, you can tell a lot about how related your observed data are and if they fit into a normal distribution.

For this submission, you will be given a series of scenarios and small collections of data. You should plot the data or calculate probabilities using excel. Then, you will create your own real or hypothetical scenario to graph and explain.

Answer the following:

  • The mean temperature for the month of July in Boston, Massachusetts is 73 degrees Fahrenheit. Plot the following data, which represent the observed mean temperature in Boston over the last 20 years:
    1998 72
    1999 69
    2000 78
    2001 70
    2002 67
    2003 74
    2004 73
    2005 65
    2006 77
    2007 71
    2008 75
    2009 68
    2010 72
    2011 77
    2012 65
    2013 79
    2014 77
    2015 78
    2016 72
    2017 74
    1. Is this a normal distribution? Explain your reasoning.
    2. What is an outlier? Are there any outliers in this distribution? Explain your reasoning fully.
    3. Using the above data, what is the probability that the mean will be over 76 in any given July?
    4. Using the above data, what is the probability that the mean will be over 80 in any given July?
  • A heatwave is defined as 3 or more days in a row with a high temperature over 90 degrees Fahrenheit. Given the following high temperatures recorded over a period of 20 days, what is the probability that there will be a heatwave in the next 10 days?
    Day 1 93
    Day 2 88
    Day 3 91
    Day 4 86
    Day 5 92
    Day 6 91
    Day 7 90
    Day 8 88
    Day 9 85
    Day 10 91
    Day 11 84
    Day 12 86
    Day 13 85
    Day 14 90
    Day 15 92
    Day 16 89
    Day 17 88
    Day 18 90
    Day 19 88
    Day 20 90

Customer surveys reveal that 40% of customers purchase products online versus in the physical store location. Suppose that this business makes 12 sales in a given day

  1. Does this situation fit the parameters for a binomial distribution? Explain why or why not?
  2. Find the probability of the 12 sales on a given day exactly 4 are made online
  3. Find the probability of the 12 sales fewer than 6 are made online
  4. Find the probability of the 12 sales more than 8 are made online

Your own example:

  • Choose a company that you have recently seen in the news because it is having some sort of problem or scandal, and complete the following:
    • Discuss the situation, and describe how the company could use distributions and probability statistics to learn more about how the scandal could affect its business.
    • If you were a business analyst for the company, what research would you want to do, and what kind of data would you want to collect to create a distribution?
    • Would this be a standard, binomial, or Poisson distribution? Why?
    • List and discuss at least 3 questions that you would want to create probabilities for (e.g., What is the chance that the company loses 10% of its customers in the next year?).
    • What would you hope to learn from calculating these probabilities?
    • Assuming that upper management does not see the value in expending the time and money necessary to collect data to analyze, make an argument (at least 100 words) convincing them that the expenditure is necessary and explaining some dangers the company could face by not knowing what the data predict.

Solutions

Expert Solution

a.

From the above histogram, it is clear that the distribution of the given data is not normal because there is no single peak at the mean value of 72.65.

b.

An outlier is an extremely lower or extremely higher value, i.e., it is that value which does not fit the given data set. There are no outliers in the given distribution because all values are within the range of 65 and 79 with no extremely lower or extremely higher values which can be observed from the above histogram.

c.

Here, X =mean temperature

Sample size, n =20 July months

The probability that the mean temperature will be over 76 in any given July =P(X > 76) =No.of July months where mean > 76/Total sample size =6/20 =0.3

d.

The probability that the mean temperature will be over 80 in any given July =P(X > 80) =No.of July months where mean > 80/Total sample size =0/20 =0


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