Estimate The Mean Of The Weights (in Pounds) Of Socks Ordered By Customers In The Last Two Hours Given In The Following Grouped Frequency Table.- Round The Final Answer To One Decimal Place.$\[ \begin{tabular}{|c|c|} \hline Value Interval &

Feb 26, 2025 by ADMIN 242 views

Estimate the Mean of the Weights of Socks Ordered by Customers

===========================================================

Introduction

In this article, we will discuss how to estimate the mean of the weights of socks ordered by customers in the last two hours given in a grouped frequency table. The mean is a measure of central tendency that represents the average value of a dataset. It is an essential statistical concept used in various fields, including mathematics, science, and engineering.

Understanding Grouped Frequency Tables

A grouped frequency table is a table that displays the frequency of data values within a range of values. It is a useful tool for summarizing large datasets and identifying patterns or trends. In this case, we are given a grouped frequency table that shows the weights of socks ordered by customers in the last two hours.

Grouped Frequency Table

Value Interval	Frequency
0.1-0.2	5
0.2-0.3	8
0.3-0.4	12
0.4-0.5	15
0.5-0.6	10
0.6-0.7	5
0.7-0.8	3
0.8-0.9	2
0.9-1.0	1

Estimating the Mean

To estimate the mean of the weights of socks ordered by customers, we need to use the formula for the mean of a grouped frequency distribution:

\bar{x} = \frac{\sum f_ix_i}{\sum f_i}

where $\bar{x}$ is the estimated mean, $f_i$ is the frequency of the $i^{th}$ interval, and $x_i$ is the midpoint of the $i^{th}$ interval.

Calculating the Midpoints

To calculate the midpoints of the intervals, we need to find the average of the upper and lower bounds of each interval.

Value Interval	Midpoint
0.1-0.2	0.15
0.2-0.3	0.25
0.3-0.4	0.35
0.4-0.5	0.45
0.5-0.6	0.55
0.6-0.7	0.65
0.7-0.8	0.75
0.8-0.9	0.85
0.9-1.0	0.95

Calculating the Sum of Frequencies

To calculate the sum of frequencies, we need to add up the frequencies of all the intervals.

\sum f_i = 5 + 8 + 12 + 15 + 10 + 5 + 3 + 2 + 1 = 61

Calculating the Sum of Products

To calculate the sum of products, we need to multiply the frequency of each interval by its midpoint and add up the results.

\sum f_ix_i = 5(0.15) + 8(0.25) + 12(0.35) + 15(0.45) + 10(0.55) + 5(0.65) + 3(0.75) + 2(0.85) + 1(0.95)

= 0.75 + 2 + 4.2 + 6.75 + 5.5 + 3.25 + 2.25 + 1.7 + 0.95 = 27.3

Calculating the Estimated Mean

Now that we have calculated the sum of frequencies and the sum of products, we can calculate the estimated mean using the formula:

\bar{x} = \frac{\sum f_ix_i}{\sum f_i} = \frac{27.3}{61} = 0.45

Conclusion

In this article, we discussed how to estimate the mean of the weights of socks ordered by customers in the last two hours given in a grouped frequency table. We used the formula for the mean of a grouped frequency distribution and calculated the midpoints of the intervals, the sum of frequencies, and the sum of products. Finally, we calculated the estimated mean using the formula.

Rounding the Final Answer

The estimated mean is 0.45. To round the final answer to one decimal place, we need to look at the digit in the tenths place, which is 5. Since 5 is greater than or equal to 5, we round up to 0.5.

The final answer is: $\boxed{0.5}$

=====================================================================================================

Q: What is the purpose of estimating the mean of the weights of socks ordered by customers?

A: The purpose of estimating the mean of the weights of socks ordered by customers is to understand the average weight of the socks ordered by customers in the last two hours. This information can be useful for businesses to make informed decisions about inventory management, pricing, and customer satisfaction.

Q: What is a grouped frequency table, and how is it used in estimating the mean?

A: A grouped frequency table is a table that displays the frequency of data values within a range of values. It is a useful tool for summarizing large datasets and identifying patterns or trends. In estimating the mean, the grouped frequency table is used to calculate the midpoints of the intervals, the sum of frequencies, and the sum of products.

Q: How do you calculate the midpoints of the intervals in a grouped frequency table?

A: To calculate the midpoints of the intervals, you need to find the average of the upper and lower bounds of each interval. For example, if the interval is 0.1-0.2, the midpoint would be (0.1 + 0.2) / 2 = 0.15.

Q: What is the formula for estimating the mean of a grouped frequency distribution?

A: The formula for estimating the mean of a grouped frequency distribution is:

\bar{x} = \frac{\sum f_ix_i}{\sum f_i}

where $\bar{x}$ is the estimated mean, $f_i$ is the frequency of the $i^{th}$ interval, and $x_i$ is the midpoint of the $i^{th}$ interval.

Q: How do you calculate the sum of frequencies in a grouped frequency table?

A: To calculate the sum of frequencies, you need to add up the frequencies of all the intervals. For example, if the grouped frequency table is:

Value Interval	Frequency
0.1-0.2	5
0.2-0.3	8
0.3-0.4	12
0.4-0.5	15
0.5-0.6	10
0.6-0.7	5
0.7-0.8	3
0.8-0.9	2
0.9-1.0	1

The sum of frequencies would be 5 + 8 + 12 + 15 + 10 + 5 + 3 + 2 + 1 = 61.

Q: How do you calculate the sum of products in a grouped frequency table?

A: To calculate the sum of products, you need to multiply the frequency of each interval by its midpoint and add up the results. For example, if the grouped frequency table is:

Value Interval	Midpoint	Frequency
0.1-0.2	0.15	5
0.2-0.3	0.25	8
0.3-0.4	0.35	12
0.4-0.5	0.45	15
0.5-0.6	0.55	10
0.6-0.7	0.65	5
0.7-0.8	0.75	3
0.8-0.9	0.85	2
0.9-1.0	0.95	1

The sum of products would be 5(0.15) + 8(0.25) + 12(0.35) + 15(0.45) + 10(0.55) + 5(0.65) + 3(0.75) + 2(0.85) + 1(0.95) = 27.3.

Q: How do you round the final answer to one decimal place?

A: To round the final answer to one decimal place, you need to look at the digit in the tenths place, which is 5. Since 5 is greater than or equal to 5, you round up to 0.5.

Q: What are some common applications of estimating the mean of a grouped frequency distribution?

A: Some common applications of estimating the mean of a grouped frequency distribution include:

Inventory management: Estimating the mean of a grouped frequency distribution can help businesses understand the average quantity of items sold or purchased, which can inform inventory management decisions.
Pricing: Estimating the mean of a grouped frequency distribution can help businesses understand the average price of items sold or purchased, which can inform pricing decisions.
Customer satisfaction: Estimating the mean of a grouped frequency distribution can help businesses understand the average level of customer satisfaction, which can inform customer service and quality control decisions.

Q: What are some common challenges associated with estimating the mean of a grouped frequency distribution?

A: Some common challenges associated with estimating the mean of a grouped frequency distribution include:

Data quality: Estimating the mean of a grouped frequency distribution requires high-quality data, which can be difficult to obtain, especially if the data is collected from multiple sources.
Data accuracy: Estimating the mean of a grouped frequency distribution requires accurate data, which can be difficult to obtain, especially if the data is collected from multiple sources.
Data interpretation: Estimating the mean of a grouped frequency distribution requires careful interpretation of the results, which can be challenging, especially if the data is complex or nuanced.