The problem is to find the mean of a grouped data set. We are given the sales intervals, relative frequencies, frequencies, and midpoints ($x$) for each interval. The following table is provided: | Sales | Relative Frequency | Frequency ($f$) | Midpoint ($x$) | |---|---|---|---| | 75-80 | 0.09 | 9 | 77.5 | | 80-85 | 0.12 | 21 | 82.5 | | 85-90 | 0.15 | 36 | 87.5 | | 90-95 | 0.11 | 47 | 92.5 | | 95-100 | 0.20 | 67 | 97.5 | | 100-105 | 0.20 | 87 | 102.5 | | 105-110 | 0.11 | 98 | 107.5 | | 110-115 | 0.2 | 108 | 112.5 |

1. Problem Description

The problem is to find the mean of a grouped data set. We are given the sales intervals, relative frequencies, frequencies, and midpoints ($x$) for each interval. The following table is provided:

| Sales | Relative Frequency | Frequency ($f$) | Midpoint ($x$) |

|---|---|---|---|

| 75-80 | 0.09 | 9 | 77.5 |

| 80-85 | 0.12 | 21 | 82.5 |

| 85-90 | 0.15 | 36 | 87.5 |

| 90-95 | 0.11 | 47 | 92.5 |

| 95-100 | 0.20 | 67 | 97.5 |

| 100-105 | 0.20 | 87 | 102.5 |

| 105-110 | 0.11 | 98 | 107.5 |

| 110-115 | 0.2 | 108 | 112.5 |

2. Solution Steps

To find the mean of the grouped data, we can use the formula:

$\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}$

where $f$ is the frequency of each interval and $x$ is the midpoint of each interval.

First, calculate $f \cdot x$ for each interval:

* 75-80: $9 \cdot 77.5 = 697.5$

* 80-85: $21 \cdot 82.5 = 1732.5$

* 85-90: $36 \cdot 87.5 = 3150$

* 90-95: $47 \cdot 92.5 = 4347.5$

* 95-100: $67 \cdot 97.5 = 6532.5$

* 100-105: $87 \cdot 102.5 = 8917.5$

* 105-110: $98 \cdot 107.5 = 10535$

* 110-115: $108 \cdot 112.5 = 12150$

Sum of $f \cdot x$:

$\sum (f \cdot x) = 697.5 + 1732.5 + 3150 + 4347.5 + 6532.5 + 8917.5 + 10535 + 12150 = 47162.5$

Next, calculate the sum of the frequencies:

$\sum f = 9 + 21 + 36 + 47 + 67 + 87 + 98 + 108 = 473$

Finally, calculate the mean:

$\text{Mean} = \frac{47162.5}{473} = 99.7093023255814 \approx 99.71$

The provided image states that $SF = 465$ and $Sfx = 47162.5$. $SF$ is incorrect, it should be 473 instead.

Using $SF=473$:

Mean = $47162.5 / 473 = 99.7093023255814$

Mean $\approx$ 99.71

3. Final Answer

The mean is approximately 99.

7

1.