Jackson Caldwell
Math 1040-014
Skittles project
Statistics and Skittles
Our statistics class at Salt Lake Community College, gathered data and use concepts that
we were taught throughout the semester are putting it to use in a term project analyzing 21
2.17oz bags of skittles showing how statistics can be used in everyday life. Topics to be explored
are categorical data, quantitative data, confidence intervals, and hypothesis tests.
Organizing and Displaying Categorical Data: Colors
The following chart shows the color and proportions of all the skittles involved.
Skittles
purple, 21.1%
red, 20.0%
red
orange
green, 19.9%
orange, 19.0%
yellow
green
purple
yellow, 20.0%
While looking at the pie and pareto charts demonstrating the class value of skittle color
proportions I found it interesting to see that some of the colors were very comparable while my
bag contained quite less green then the rest of the class. I had expected that the skittle colors
would be more evenly distributed but I was wrong. Although my bag had fewer green skittles
than most, it made up for the deficit in all the other colors.
Pareto Chart Total
270
265
260
255
250
245
240
235
230
225
Sum of
purple
Sum of red
Sum of
yellow
Sum of green
Sum of
orange
My bag of Skittles:
Total number of candies in sample = 61
Number of
red candies
Number of
orange
candies
Number of
yellow
candies
Number of
green candies
Number of
purple
candies
14
12
13
7
15
The total number of candies in the sample = 1252
The entire sample:
Number of
red candies
Number of
orange
candies
Number of
yellow
candies
Number of
green
candies
Number of
purple
candies
251
238
250
249
264
Proportion .200
.190
.200
.199
.211
Organizing and Displaying Quantitative Data: the number of candies per bag
The total number of candies in your own single 2.17-ounce bag of Skittles = 61
The total number of bags in the sample collected by the entire class = 21
The total number of candies in the sample collected by the entire class =1252
For the entire sample:
𝑥̅ = 59.6 (the mean number of candies per bag rounded to 1 decimal place)
s = 2.75 (the std. deviation of the number of candies per bag rounded to two decimal places)
Candies Per Bag Frequency Histogram
10
9
Frequency
8
9
6
4
2
1
0
51-53
54-56
2
0
57-59
60-62
63-65
Skittles Per Bag
5- number summary: (round to one decimal place where necessary)
Min: 51
51
Q1: 58
Median: 60
58
Q3: 61.5
60
61.5
Max: 64
64
While organizing and displaying quantitative data I noticed that the shape of the
histogram was almost in a normal distribution. What causes the graph to not be a normal
distribution is the gap created between 54-56 where no one in our class had a pack of skittles
containing this number. The graphs did reflect what I expected to see and the data of the whole
class does agree with mine. Also, the box plot shows the mean which was very similar to the
number of candies in my bag of skittles.
Reflection: Categorical Versus Quantitative Data
To this point I have presented two types of data, categorical and quantitative. Categorical,
also known as qualitative data, consists of names or labels not numbers or measurements. One
example in our skittles data is the colors of the skittles. Charts used to represent categorical data
include pie charts and pareto charts as seen before. Charts that wouldn’t make sense representing
categorical data include box plots, frequency histograms, histograms, dotplots, and scatterplots
because these all involve numbers not names or categories. Categorical data does not involve
calculations. This data is used when shedding light upon certain data, like the average proportion
of color in skittles. Quantitative, or numerical data, consist of numbers that represent numbers or
measurements. Examples from the skittle data shown above are frequency histograms and
boxplots. Other examples used for numerical data include stem and leaf plot, line graphs,
scatterplots, and time series graphs. Graphs that wouldn’t make sense as numerical graphs are
any graphs that talk about specific categories not involving numbers. Numerical data has
numerous calculations. The main ones used in the skittles project, up to this point, include the
mean, standard deviation, and the five number summary. As we continue to analyze the skittle
data other numerical calculations will be shown. The importance of these calculations let us look
deeper into statistics to make better inferences and conclusions.
Confidence Interval Estimates
A confidence interval is a range of values used to estimate the true value of a population
parameter. A population parameter includes the proportion of a population, the mean of a
population, and the standard deviation of a population. With a confidence interval also comes a
confidence level, the probability that 1- alpha (such as .95) that the confidence interval does
contain the population parameter, population mean, or the standard deviation of a population.
Below are confidence intervals, one for the proportion, one for the mean, and one for the
standard deviation. The following given decimals are the probability of the accuracy of the given
interval. After constructing a 95% confidence interval estimate for the true proportion of purple
candies I discovered that:
.188 < p < .233.
This statistic is 95 percent confident that this is the true proportion of purple candies per bag of
2.17-ounce skittles.
After solving a 99% confidence interval for the true mean of the number of candies per bag of
skittles I found that mean is:
57.893 <
u
< 61.307
This confidence inteval is 99% confident that all 2.17-ounce bags of skittles ever packaged
contain a mean listed in the above interval.
After formulating another confidence interval of 98% for the standard deviation in a 2.17-ounce
bag of skittles I concluded that:
2.007 <
o
< 4.279
This confidence interval is 98% confident to hold the value of standard deviation of the number
of skittles in every 2.17-ounce bag.
See notes at end of paper to see calcultations.
Hypothesis Tests
In general a hypothesis is an educated guess about the explaination to a scientific
question. In statistics a hypothesis has a little bit of a different meaning. Statistically a hypothesis
is a claim or statement about a property of a population. A hypothesis test is a procedure for
proven, a test never tries to prove the hypothesis it only fails to disprove it. Several components
make up a hypothesis test, including the null and alternative hypothesis, the test stastic, the
finding of the p-value or critical value, and the conclusion regarding a claim. Below are two
claims to be tested, one about the amount of green skittles in a 2.17-ounce bag and another about
the mean number of candies in a bag of skittles.
Claim: 20% of skittle candies are green.
See notes at the end of paper for calculations
Conclusion: Fail to reject the null. There wasn’t sufficient enough evidence to warrant a rejection
of the claim that 20% of skittles are green. The p-value is greater than the alpha value there for it
fails to reject the null.
Claim: The mean number of candies in a bag or skittles is 56.
See notes at end of paper for calculations.
Conclusion: Fail to reject the null hypothesis. There wasn’t sufficient enough evidence to
warrant a rejection of the claim that the mean of skittles per bag is 56. The p-value is
substantually bigger than the alpha value leading to a failure of rejection.
Confidence Interval and Hypothesis Testing Conclusion
Several conditions need to be met for these different calculations to be sucessful. For
confidence intervals for the mean the sample must be a simple random sample and be normally
distributed. The same goes for the population proportion and the standard deviation. One
exception for the proportion is that if the sample isn’t normally distrubuted or that information is
not given one can continue with their calculations if n>30. For hypothesis testing of a population
proportion the requirements are similar to the confidence interval requirements listed above. The
only difference is that the binomial distribution must also be satisfied, as well as meau equals np
and sigma equals npq. The conditions for the hypothesis testing of a claim about the mean are the
same as the confidence interval but if n> 30 this also satisfies the requirement of normal
distribution. While working with skittles and performing certain calcultations I noticed that all of
these requirements were met. If they failed to do so calculations could not be taken futher.
This statistical project could be improved by having each student bring in their 2.17ounce bags of skittles to class to ensure that proper counting and proper reporting is done. Errors
could have been made by counting a candy twice or not writing the correct number down when
reporting colors. The conslusions that I have drawn are that 20% of skittles are green and that the
mean of a bag of skittles can be claimed 56 according to the hypothesis test done above.