Question

1. you are going to start your own side business of completing yard work. you research lawn mowers online and at hardware stores and found the following prices: $150, $100, $128, $160, $300, $160, $175

sigma notation: ____________
mean: ____________
determine if the data set is bimodal.

Explanation:

Sigma Notation:

Step1: Identify the data set

We have the prices: \( 150, 100, 128, 160, 300, 160, 175 \). Let \( x_i \) represent the \( i \)-th price, where \( i \) ranges from 1 to 7 (since there are 7 data points).

Step2: Write the sigma notation

The sum of these prices in sigma notation is \( \sum_{i = 1}^{7} x_i \), where \( x_1 = 150 \), \( x_2 = 100 \), \( x_3 = 128 \), \( x_4 = 160 \), \( x_5 = 300 \), \( x_6 = 160 \), \( x_7 = 175 \).

Mean:

Step1: Find the sum of the data

First, we calculate the sum of the prices. Using the sigma notation result, we sum the values: \( 150 + 100 + 128 + 160 + 300 + 160 + 175 \).
\[

$$\begin{align*} &150+100 = 250\\ &250 + 128 = 378\\ &378+160 = 538\\ &538 + 300 = 838\\ &838+160 = 998\\ &998+175 = 1173 \end{align*}$$

\]

Step2: Calculate the mean

The mean (average) is the sum of the data divided by the number of data points. There are 7 data points. So the mean \( \bar{x}=\frac{\sum_{i = 1}^{7} x_i}{7} \). Substituting the sum we found: \( \bar{x}=\frac{1173}{7}=167.5714\cdots\approx 167.57 \) (rounded to two decimal places) or as a fraction \( \frac{1173}{7} \).

Sigma Notation Answer: \( \boldsymbol{\sum_{i = 1}^{7} x_i} \) (where \( x_1 = 150, x_2 = 100, x_3 = 128, x_4 = 160, x_5 = 300, x_6 = 160, x_7 = 175 \))

Mean Answer: \( \boldsymbol{\frac{1173}{7}\approx 167.57} \) (or \( \boldsymbol{167.57} \) if rounded to two decimal places)

Answer:

Step1: Find the sum of the data

First, we calculate the sum of the prices. Using the sigma notation result, we sum the values: \( 150 + 100 + 128 + 160 + 300 + 160 + 175 \).
\[

$$\begin{align*} &150+100 = 250\\ &250 + 128 = 378\\ &378+160 = 538\\ &538 + 300 = 838\\ &838+160 = 998\\ &998+175 = 1173 \end{align*}$$

\]

Step2: Calculate the mean

The mean (average) is the sum of the data divided by the number of data points. There are 7 data points. So the mean \( \bar{x}=\frac{\sum_{i = 1}^{7} x_i}{7} \). Substituting the sum we found: \( \bar{x}=\frac{1173}{7}=167.5714\cdots\approx 167.57 \) (rounded to two decimal places) or as a fraction \( \frac{1173}{7} \).

Sigma Notation Answer: \( \boldsymbol{\sum_{i = 1}^{7} x_i} \) (where \( x_1 = 150, x_2 = 100, x_3 = 128, x_4 = 160, x_5 = 300, x_6 = 160, x_7 = 175 \))

Mean Answer: \( \boldsymbol{\frac{1173}{7}\approx 167.57} \) (or \( \boldsymbol{167.57} \) if rounded to two decimal places)