Formula average
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1010ReviewsProject.docxA Review of Averaging Formulas
Math 1010 Intermediate Algebra Group Project
The Story:
Alyssa works for an online company that helps thousands of small business owners across the country to advertise themselves and find customers in need of their services.
After the job is completed, Alyssa’s company encourages the customers to review the business on a scale of 1 to 5 stars. This review is saved within Alyssa’s database so that future customers can see the average number of stars a business has received on the online service. That way the customers can find the best rated person for the job.
Alyssa found that many business owners had questions about how they could improve their rating. One common question was: “How many 5-star reviews in a row do I need to improve my rating from where I’m at to a 4.5?” Or a 4.2? Or whatever threshold they were interested in? Rather than having to work this out from scratch in every scenario, it is more efficient to create a formula so the answer can be found quickly and efficiently.
We will start out with some simpler questions to become familiar with the situation and how averages work. Then we will work our way towards the full formula to help the business owners answer their question above. You will use skills you learned about solving formulas for a particular variable and working with rational expressions.
AVERAGES
We will be exploring how basic averages work in this section. This Average is also called the Mean, or Arithmetic Mean.
1. Write a formula which can find the average of two numbers x and y.
AVERAGE =
2. Use your formula to find the average of 16 and 34. Show your process.
AVERAGE =
3. Explain in words: How does the process of finding the average change if there are 6 numbers to average?
4. Find the average of the six numbers 4, 6, 7, 12, 14, and 17. Show your process.
FREQUENCY TABLES
Often very large data sets are averaged. The data sets include many numbers which are the same. A Frequency Table describes how many of each number are included in the set. Examine the following table as an example:
|
Number |
Frequency |
|
1 |
2 |
|
2 |
4 |
|
3 |
3 |
The table explains the data set 1, 1, 2, 2, 2, 2, 3, 3, 3.
5. Find the average of the nine numbers above. Show your process. Round your answer to the nearest hundredth.
When the data set becomes too large, it becomes too tedious to add all of the numbers one by one. We need a different strategy. Examine the next frequency table, which shows the number of reviews a particular business owner has received of each number of stars.
|
Stars |
Frequency |
|
1 |
21 |
|
2 |
11 |
|
3 |
12 |
|
4 |
46 |
|
5 |
60 |
6. Determine how many total reviews this business has received. Show your process.
7. Rather than add up all of those numbers one at a time, we will add them in groups. As an example, if we add up all twelve of the 3s, we should use multiplication (since it represents adding the same number repeatedly). All of the 3s sum to 3 times 12 which is 36. In the following blanks, similarly write the sum of all numbers of each type.
Sum of 1s __________
Sum of 2s __________
Sum of 3s ___36___
Sum of 4s __________
Sum of 5s __________
8. Use your results from the previous two parts to find the average number of stars the business has received. Show your process. Round your answer to the nearest hundredth.
ONE MORE NUMBER
To answer our final question, we will need to know what happens to an average if we add one more number.
9. Suppose a list of seven numbers has an average of 12. What is the SUM of those seven numbers? Show your process.
10. Suppose the number 18 is added to the previous list of seven numbers to make eight numbers. What is the new SUM of all eight numbers?
11. What is the new average of all eight numbers? Show your process.
MANY MORE REVIEWS
12. Suppose a business has a current average review of 3.2 stars. This is from 30 customer reviews. Using a similar process to the previous three questions, find the new average review for the business after four new 5-star reviews. Show your process and round your answer to the nearest hundredth.
PUTTING IT ALL TOGETHER
Suppose a business currently has an average rating of R stars. This average is calculated from N current reviews.
14. If the business wants to know how many 5-star reviews in a row are needed to achieve a desired new rating, the formula in the previous part must be solved for X. Solve your formula in the previous part for X, and show your process.
Let’s put our new formula into action! Kevin’s Tree Service is a small business that currently has ratings given in the table below:
|
Stars |
Frequency |
|
1 |
4 |
|
2 |
1 |
|
3 |
10 |
|
4 |
11 |
|
5 |
18 |
15. First calculate the current average review for Kevin’s Tree Service. Show your process. Leave your answer in the form of a fraction.
AVERAGE =
16. Now use your formula from Part 14 to calculate how many 5-star reviews in a row are needed for Kevin’s Tree Service to achieve an average review of 4.5.
REFLECTION
17. Did this project change the way you think about how math can be applied in the real world? Do you believe this kind of analysis could be important to business owners? Did you expect Kevin’s Tree Service to need that many 5-star reviews to reach its goal? Does what you learned here make you more likely to carefully consider ratings you give for services you receive? Write at least two paragraphs addressing the above questions. Refer to specific parts of this project where
