Find The Weighted Mean Of The Percentages Of People Who Liked The Taste Of A New Diet Cola. Round To The Nearest Tenth Of A Percent.$[ \begin{tabular}{ccc} Area & % Favored & Number Surveyed \ \hline 1 & 45 & 2400 \ 2 & 75 & 3100 \ 3 & 35 &

by ADMIN 241 views

Introduction

In the world of statistics, calculating the weighted mean is a crucial concept that helps us understand the average value of a dataset when each data point has a different weight or importance. In this article, we will delve into the world of weighted means and apply this concept to a real-world scenario: finding the weighted mean of the percentages of people who liked the taste of a new diet cola.

Understanding Weighted Mean

The weighted mean is a type of average that takes into account the relative importance or weight of each data point. It is calculated by multiplying each data point by its corresponding weight and then summing up the results. The formula for calculating the weighted mean is:

Weighted Mean = (Σ (x_i * w_i)) / Σ w_i

where x_i is the data point, w_i is the weight, and Σ denotes the sum.

The Diet Cola Taste Preference Survey

A survey was conducted to gather data on the taste preferences of people who tried a new diet cola. The survey consisted of three areas, each with a different percentage of people who favored the taste of the diet cola. The data is presented in the following table:

Area % Favored Number Surveyed
1 45 2400
2 75 3100
3 35 4600

Calculating the Weighted Mean

To calculate the weighted mean, we need to multiply each percentage by the number surveyed and then sum up the results. We will also need to calculate the total number surveyed, which is the sum of the number surveyed in each area.

Total Number Surveyed = 2400 + 3100 + 4600 = 10100

Now, let's calculate the weighted sum:

Weighted Sum = (45 * 2400) + (75 * 3100) + (35 * 4600) = 108000 + 232500 + 161000 = 501500

Calculating the Weighted Mean

Now that we have the weighted sum, we can calculate the weighted mean by dividing the weighted sum by the total number surveyed.

Weighted Mean = Weighted Sum / Total Number Surveyed = 501500 / 10100 = 49.7

Rounding to the Nearest Tenth of a Percent

The weighted mean is 49.7, which is already rounded to the nearest tenth of a percent. Therefore, the weighted mean of the percentages of people who liked the taste of the new diet cola is 49.7%.

Conclusion

In this article, we applied the concept of weighted mean to a real-world scenario: finding the weighted mean of the percentages of people who liked the taste of a new diet cola. We calculated the weighted mean by multiplying each percentage by the number surveyed and then summing up the results. The weighted mean was found to be 49.7%, which is the average value of the dataset when each data point has a different weight or importance.

Discussion

The weighted mean is a powerful tool in statistics that helps us understand the average value of a dataset when each data point has a different weight or importance. In this article, we applied this concept to a real-world scenario and found the weighted mean of the percentages of people who liked the taste of a new diet cola. The weighted mean was found to be 49.7%, which is the average value of the dataset when each data point has a different weight or importance.

Limitations

One limitation of this study is that it assumes that the data is normally distributed. However, in real-world scenarios, the data may not always be normally distributed. Therefore, it is essential to check the distribution of the data before applying the weighted mean.

Future Directions

In future studies, it would be interesting to explore other applications of the weighted mean, such as in finance or economics. The weighted mean can be used to calculate the average return on investment or the average growth rate of a company.

References

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of sources.

Introduction

In our previous article, we explored the concept of weighted mean and applied it to a real-world scenario: finding the weighted mean of the percentages of people who liked the taste of a new diet cola. In this article, we will answer some frequently asked questions about weighted mean to help you better understand this concept.

Q: What is the weighted mean?

A: The weighted mean is a type of average that takes into account the relative importance or weight of each data point. It is calculated by multiplying each data point by its corresponding weight and then summing up the results.

Q: How is the weighted mean different from the arithmetic mean?

A: The weighted mean is different from the arithmetic mean in that it takes into account the relative importance or weight of each data point. The arithmetic mean, on the other hand, gives equal weight to each data point.

Q: When should I use the weighted mean?

A: You should use the weighted mean when you have a dataset with different weights or importance for each data point. For example, in a survey where some respondents are more representative of the population than others, you would use the weighted mean to calculate the average response.

Q: How do I calculate the weighted mean?

A: To calculate the weighted mean, you need to multiply each data point by its corresponding weight and then sum up the results. The formula for calculating the weighted mean is:

Weighted Mean = (Σ (x_i * w_i)) / Σ w_i

where x_i is the data point, w_i is the weight, and Σ denotes the sum.

Q: What are some common applications of the weighted mean?

A: The weighted mean has many applications in various fields, including:

  • Finance: to calculate the average return on investment or the average growth rate of a company
  • Economics: to calculate the average GDP or the average inflation rate
  • Statistics: to calculate the average value of a dataset with different weights or importance for each data point
  • Marketing: to calculate the average response rate of a survey or the average conversion rate of a marketing campaign

Q: What are some common mistakes to avoid when calculating the weighted mean?

A: Some common mistakes to avoid when calculating the weighted mean include:

  • Not using the correct weights or importance for each data point
  • Not summing up the results correctly
  • Not checking the distribution of the data before applying the weighted mean

Q: How do I check the distribution of the data before applying the weighted mean?

A: To check the distribution of the data, you can use statistical tests such as the normality test or the Shapiro-Wilk test. If the data is not normally distributed, you may need to use a different type of average, such as the median or the mode.

Q: What are some common tools or software used to calculate the weighted mean?

A: Some common tools or software used to calculate the weighted mean include:

  • Microsoft Excel: you can use the formula =AVERAGE(range, weights) to calculate the weighted mean
  • Google Sheets: you can use the formula =AVERAGE(range, weights) to calculate the weighted mean
  • R: you can use the function weighted.mean() to calculate the weighted mean
  • Python: you can use the function numpy.average() to calculate the weighted mean

Conclusion

In this article, we answered some frequently asked questions about weighted mean to help you better understand this concept. We hope that this article has been helpful in clarifying any doubts you may have had about weighted mean.

References