Question

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.

Answer 1

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