Question

Assignment Details

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 the

Answer 1

Sol:

(a)

Mean:72.65

Median: 72.5

Mode: 72.77

Normal Distribution is a symmetric distribution. With this said, the mean, median and the mode are all the same for a normal distribution.

The equality of the mean, median and mode makes a normal distribution curve to be a Bell curve . This is the graphical representation of a normal distribution curve :-

Here in the graph you can see :-

• Mean = Median = Mode.
• Symmetry in the center. This means that the distribution is symmetrically skewed.
• 50% of the data is to the left to the mean and remaining 50% is to the right of the mean.
• And of course, it is a Bell shaped curve.

(b)

In statistics, an outlier is an observation point that is distant from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the data set.

Outlier detected?	No
Significance level:	0.05 (two-sided)
Critical value of Z:	2.70824545658

Your data

Row	Value	Z	Significant Outlier?
1	72.	0.15
2	69.	0.84
3	78.	1.23
4	70.	0.61
5	67.	1.30
6	74.	0.31
7	73.	0.08
8	65.	1.76	Furthest from the rest, but not a significant outlier (P > 0.05).
9	77.	1.00
10	71.	0.38
11	75.	0.54
12	68.	1.07
13	72.	0.15
14	77.	1.00
15	65.	1.76
16	79.	1.46
17	77.	1.00
18	78.	1.23
19	72.	0.15
20	74.	0.31

(c)

Sample Standard Deviation, s

4.34408

Mean=72.65

Since we are asked about the mean (not an individual observation), we need to calculate the standard deviation for the mean: ?x = 4.34408 / ? 20 = 0.9714.

Now, we calculate the z-score, z = 76?72.65 / 0.9714 = 3.4486

The P-Value is 0.000282.

(d)

Since we are asked about the mean (not an individual observation), we need to calculate the standard deviation for the mean: ?x = 4.34408 / ? 20 = 0.9714.

Now, we calculate the z-score, z = 80?72.65 / 0.9714 = 7.566

The P-Value is < 0.00001

This number is larger than the z-scores on the table, so we can assume that the table would give us (approximately) 1. Since we are asked about “greater than,” we need to subtract this probability from 1, so we get 1 ? 1 = 0.