Question

The following data was collected from 1 bag of Hershey Kisses®. Each Kiss® was weighed (in grams) and recorded in the table below. Hershey claims that there is 368 grams of chocolate in one bag.

4.76	4.72	4.74	4.55	4.91	4.74	4.78	4.71	4.8
4.78	4.78	4.75	4.79	4.82	4.91	4.83	4.68	4.74
4.7	4.8	4.7	4.76	4.7	4.83	4.93	4.74	4.84
4.82	4.78	4.77	4.72	4.78	4.83	4.75	4.74	4.68
4.84	4.71	4.71	4.76	4.66	4.78	4.73	4.74	4.92
4.77	4.8	4.79	4.86	4.64	4.78	4.7	4.75	4.78
4.76	4.83	4.66	4.77	4.83	4.78	4.69	4.81	4.68
4.78	4.88	4.72	4.85	4.85	4.81	4.74	4.8	4.82
4.84	4.7	4.85	4.7	4.81	4.72	4.79	4.63

To help you answer the questions below use your scientific calculator. Your scientific calculator is capable of doing calculations on entire data sets by first entering the data and then pressing combinations of keys to find the average and standard deviation etc... You should check with your calculator manual to see how this special data handling feature works. Let the instructor know if you have any questions. You will need to learn how to do this for testing purposes. Note: Instructions for several brands of calculators in included in the folder Course Overview/Excel & Calculator Instructions.

1. What is the Mean and Median? (you may want to use your calculator!)

2. In general, each Kiss® is approximately how many grams? Explain what measure you used and why.

3. What is the Range? Are you surprised at this? Why or why not?

4. What could be some reasons for variation in the weights of the Kisses®? NOTE: Take time answering this one. There are lots of thingsto consider here and I'll be looking for a well thought out answer with several given reasons contributing to the variation. Of course, the wrappers and tags could vary but what about the drops of chocolate themselves? Why aren't they all the same?

5. Would you say that there are any two Kisses that could have exactly the same weight? (I mean exactly the same weight!)

6. How many Kisses® were there in the bag?

7. Based on Hersheys® claim for 368 total net grams of chocolate in the bag, approximately how many Kisses® too many or too few are there? Give some possible explanations for this difference.

8. EXCEL: Click on and print out one of the following: Excel Descriptive Statistics 2016/2013 to see how to enter the Kiss data into a worksheet and obtain a list of descriptive statistics and a histogram with no more than 12 classes. Also, make sure to sort your data using the Sort command under Data on the menu bar. Submit your Excel file to the Lab1 Part 1 Dropbox.

9. Standard Deviation & Empirical Rule: Variation is a big factor in the analysis of most any data set and it will be very important to have a way of measuring it. Standard Deviation is one such measure that you will study and learn to calculate in an upcoming section. For now, find the Standard Deviation number on your Descriptive Statistics read-out from Excel. There is a rule for "mound-shaped" distributions that can help you have some feeling for what this standard deviation number is telling you. It's called the Empirical Rule and is stated below:
For any data set having a bell-shaped (or mound-shaped) distribution the following are true:
- Approximately 68% of the data values will be within one standard deviation of the mean.
- Approximately 95% of the data values will be within two standard deviation of the mean.
- Almost all of the data values will be within three standard deviation of the mean.
Use the standard deviation value given in Excel and the Empirical Rule (stated above) to find answers to the following:
a) Find the percentage of all the Kisses in the bag that fell within 1 standard deviation of the mean? ... within 2?, … within 3?
(Show how you calculated these percentages!)
b) How close is the Empirical Rule in predicting the percentages that you calculated above?
c) If your calculated percentages did not line up with the percentages claimed by the Empirical Rule, speculate on some possible reasons for this.

10. How might standard deviation and the shape of the distribution indicate how consistent Hershey® is in the manufacturing of their Kisses®?

Answer 1

Answer:

Based on the given data:

1. What is the Mean and Median?

Mean of the sample can be calculated using the formula:

• 
• Excel formula for mean: AVERAGE(number1, number2,....)

Median of the sample (even number of observations n = 80) can be calculated as below:

• First organize the data in ascending order:
• median (even count) =
• Excel formula: MEDIAN(number1, number2, ...)

2. In general, each Kiss® is approximately how many grams? Explain what measure you used and why.

• Each kiss is around 4.77 grams, since the distribution appears symmetric, therefore mean can be used as a measure of central tendency.

3. What is the Range? Are you surprised at this? Why or why not?

• Range is the difference between the lowest and the highest value
• Excel formula, lowest value = MIN(number1, number2, ...) = 4.55
• Excel formula, highest value = MAX(number1, number2,...) = 4.93
• Range = 4.93 - 4.55 = 0.38

4. What could be some reasons for variation in the weights of the Kisses®? NOTE: Take time answering this one. There are lots of things to consider here and I'll be looking for a well thought out answer with several given reasons contributing to the variation. Of course, the wrappers and tags could vary but what about the drops of chocolate themselves? Why aren't they all the same?

• Variation could be due to different attributes like variation due to machines used in the manufacturing process.
• Variability during packaging.

5. Would you say that there are any two Kisses that could have exactly the same weight? (I mean exactly the same weight!)

• Weight is a continous random variable and there will be infinitesimal values of weight in any given interval. Two kisses could not have exactly the same weight.

6. How many Kisses® were there in the bag?

• The sample provided consisted of 80 Hershey Kisses in one bag.

7. Based on Hersheys® claim for 368 total net grams of chocolate in the bag, approximately how many Kisses® too many or too few are there? Give some possible explanations for this difference.

• Weight of the total Hershey kisses in the bag can be determine using the sum formula in excel
• Total sum of 80 Hersey Kisses in bag = SUM(number1, number2....) = 381.48 = ~381.5 g
• This is 381.5 - 368 = 13.48 grams excess than the expected weight of one bag
• Number of extra Hershey Kisses in the bag = Extra weight / Average weight of 1 Hershey Kiss = 13.48/4.7685 = 2.83 or ~ 3 extra Hershey Kisses

8. EXCEL: Click on and print out one of the following: Excel Descriptive Statistics 2016/2013 to see how to enter the Kiss data into a worksheet and obtain a list of descriptive statistics and a histogram with no more than 12 classes. Also, make sure to sort your data using the Sort command under Data on the menu bar. Submit your Excel file to the Lab1 Part 1 Dropbox.

• Sort the data in ascending order
• Choose Data --> Data Analysis --> Descriptive Statistics ---> Summary Statistics

• Mean	4.7685
  Standard Error	0.007722
  Median	4.775
  Mode	4.78
  Standard Deviation	0.069064
  Sample Variance	0.00477
  Kurtosis	0.584048
  Skewness	-0.13273
  Range	0.38
  Minimum	4.55
  Maximum	4.93
  Sum	381.48
  Count	80

For Histogram: Choose Data --> Data Analysis --> Histogram ---> Check Labels and Chart Output

Class	Frequency
4.55	0
4.6	0
4.65	2
4.7	12
4.75	18
4.8	24
4.85	17
4.9	2
4.95	4
5	0
More	0

9. Standard Deviation & Empirical Rule: Variation is a big factor in the analysis of most any data set and it will be very important to have a way of measuring it. Standard Deviation is one such measure that you will study and learn to calculate in an upcoming section. For now, find the Standard Deviation number on your Descriptive Statistics read-out from Excel. There is a rule for "mound-shaped" distributions that can help you have some feeling for what this standard deviation number is telling you. It's called the Empirical Rule and is stated below:
For any data set having a bell-shaped (or mound-shaped) distribution the following are true:
- Approximately 68% of the data values will be within one standard deviation of the mean.
- Approximately 95% of the data values will be within two standard deviation of the mean.
- Almost all of the data values will be within three standard deviation of the mean.
Use the standard deviation value given in Excel and the Empirical Rule (stated above) to find answers to the following:
a) Find the percentage of all the Kisses in the bag that fell within 1 standard deviation of the mean? ... within 2?, … within 3?
(Show how you calculated these percentages!)
b) How close is the Empirical Rule in predicting the percentages that you calculated above?
c) If your calculated percentages did not line up with the percentages claimed by the Empirical Rule, speculate on some possible reasons for this.

10. How might standard deviation and the shape of the distribution indicate how consistent Hershey® is in the manufacturing of their Kisses®?