Question

a) construct a frequency distribution for the data.
b) use the frequency distribution to determine the mean.
c) use the frequency distribution to determine the median and mode.
d) write the formula for the mean in sigma notation and use it to calculate the mean cell phone case price. round your answer to the nearest cent.

price	frequency
19.99	3
29.99	3
35.99	5
39.99	1
49.99	5
54.99	1
79.99	1
99.99	1

handwritten calculations: (\bar{x} = \frac{sum (x_i cdot f_i)}{22}), (\bar{x} = (8.99 + 19.99 + 29.99 + 35.99 + 39.99 + 49.99 + 54.99 + 79.99 + 99.99) / 22 = 41.17)

Explanation:

Part (d)

Step 1: Recall the formula for the mean using sigma notation

The formula for the mean \(\bar{x}\) of a frequency - distribution is \(\bar{x}=\frac{\sum_{i = 1}^{n}(x_{i}\cdot f_{i})}{\sum_{i = 1}^{n}f_{i}}\), where \(x_{i}\) is the value of the \(i\) - th class (or data point), \(f_{i}\) is the frequency of the \(i\) - th class, and \(n\) is the number of classes (or data - point groups). First, we need to calculate \(\sum_{i = 1}^{n}(x_{i}\cdot f_{i})\) and \(\sum_{i = 1}^{n}f_{i}\).

Step 2: Calculate \(\sum_{i = 1}^{n}f_{i}\)

We sum up all the frequencies: \(f_1 = 2,f_2=3,f_3 = 3,f_4=5,f_5 = 1,f_6=5,f_7 = 1,f_8=1,f_9 = 1\)
\(\sum_{i = 1}^{9}f_{i}=2 + 3+3 + 5+1+5+1+1+1=22\)

Step 3: Calculate \(\sum_{i = 1}^{n}(x_{i}\cdot f_{i})\)

We calculate \(x_{i}\cdot f_{i}\) for each row:

• For \(x_1 = 8.99\) and \(f_1 = 2\): \(x_1\cdot f_1=8.99\times2 = 17.98\)
• For \(x_2 = 19.99\) and \(f_2 = 3\): \(x_2\cdot f_2=19.99\times3=59.97\)
• For \(x_3 = 29.99\) and \(f_3 = 3\): \(x_3\cdot f_3=29.99\times3 = 89.97\)
• For \(x_4 = 35.99\) and \(f_4 = 5\): \(x_4\cdot f_4=35.99\times5=179.95\)
• For \(x_5 = 39.99\) and \(f_5 = 1\): \(x_5\cdot f_5=39.99\times1 = 39.99\)
• For \(x_6 = 49.99\) and \(f_6 = 5\): \(x_6\cdot f_6=49.99\times5 = 249.95\)
• For \(x_7 = 54.99\) and \(f_7 = 1\): \(x_7\cdot f_7=54.99\times1=54.99\)
• For \(x_8 = 79.99\) and \(f_8 = 1\): \(x_8\cdot f_8=79.99\times1 = 79.99\)
• For \(x_9 = 99.99\) and \(f_9 = 1\): \(x_9\cdot f_9=99.99\times1=99.99\)

Now, sum up these products:
\[

$$\begin{align*} \sum_{i = 1}^{9}(x_{i}\cdot f_{i})&=17.98+59.97 + 89.97+179.95+39.99+249.95+54.99+79.99+99.99\\ &=(17.98+59.97)+(89.97+179.95)+(39.99+249.95)+(54.99+79.99)+99.99\\ &=77.95+269.92+289.94+134.98+99.99\\ &=(77.95+269.92)+(289.94+134.98)+99.99\\ &=347.87+424.92+99.99\\ &=772.79+99.99\\ &=872.78 \end{align*}$$

\]

Step 4: Calculate the mean

Using the formula \(\bar{x}=\frac{\sum_{i = 1}^{n}(x_{i}\cdot f_{i})}{\sum_{i = 1}^{n}f_{i}}\), we substitute \(\sum_{i = 1}^{n}(x_{i}\cdot f_{i}) = 872.78\) and \(\sum_{i = 1}^{n}f_{i}=22\)
\(\bar{x}=\frac{872.78}{22}\approx39.67\) (rounded to the nearest cent)

Answer:

The formula for the mean in sigma notation is \(\bar{x}=\frac{\sum_{i = 1}^{n}(x_{i}\cdot f_{i})}{\sum_{i = 1}^{n}f_{i}}\), and the mean cell phone case price is approximately \(\$39.67\)