Answer :
Alright, let's analyze the waiting times recorded for these 19 customers at Ellerton Bank. We will calculate three key statistical measures: the mean, median, and standard deviation. I'll walk you through each of these calculations step-by-step.
### Step 1: List the Waiting Times
First, let's list the 19 waiting times:
[tex]\( 1, 3, 4, 5, 5, 6, 6, 7, 9, 9, 11, 12, 14, 15, 20, 22, 25, 30, 38 \)[/tex]
### Step 2: Calculate the Mean
The mean (average) waiting time is calculated by summing up all the waiting times and then dividing by the number of data points (19).
[tex]\[ \text{Mean} = \frac{\sum x}{n} \][/tex]
Where [tex]\( \sum x \)[/tex] is the sum of all waiting times, and [tex]\( n \)[/tex] is the number of waiting times.
[tex]\[ \sum x = 1 + 3 + 4 + 5 + 5 + 6 + 6 + 7 + 9 + 9 + 11 + 12 + 14 + 15 + 20 + 22 + 25 + 30 + 38 = 242 \][/tex]
[tex]\[ \text{Mean} = \frac{242}{19} \approx 12.736842105263158 \][/tex]
Thus, the mean waiting time is approximately [tex]\( 12.74 \)[/tex] minutes.
### Step 3: Calculate the Median
The median is the middle number in a sorted list of numbers. Since we have an odd number of waiting times (19), the median is the value at position [tex]\((n + 1) / 2 \)[/tex] in the ordered list.
The sorted waiting times are already given in order:
[tex]\[ 1, 3, 4, 5, 5, 6, 6, 7, 9, 9, 11, 12, 14, 15, 20, 22, 25, 30, 38 \][/tex]
The middle position is:
[tex]\((19 + 1) / 2 = 10 \)[/tex]
So, the median is the 10th number in the sorted list, which is [tex]\( 9 \)[/tex].
### Step 4: Calculate the Standard Deviation
The standard deviation measures the amount of variation or dispersion of a set of values. The formula for the standard deviation [tex]\( \sigma \)[/tex] is:
[tex]\[ \sigma = \sqrt{\frac{1}{n} \sum (x_i - \mu)^2} \][/tex]
Where:
- [tex]\( x_i \)[/tex] are the individual waiting times
- [tex]\( \mu \)[/tex] is the mean waiting time
- [tex]\( n \)[/tex] is the number of data points
We have already determined [tex]\( \mu \)[/tex] (the mean) to be approximately 12.74. After substituting the given waiting times and the calculated mean:
[tex]\[ \begin{aligned} \sigma &= \sqrt{\frac{1}{19} [(1-12.74)^2 + (3-12.74)^2 + \ldots + (38-12.74)^2]} \\ &\approx 9.775598542790416 \end{aligned} \][/tex]
Thus, the standard deviation is approximately [tex]\( 9.78 \)[/tex] minutes.
### Summary
- Mean: [tex]\( 12.74 \)[/tex] minutes
- Median: [tex]\( 9 \)[/tex] minutes
- Standard Deviation: [tex]\( 9.78 \)[/tex] minutes
These statistical measures provide valuable insights into the waiting times at Ellerton Bank, indicating the average time customers wait, the middle value of the waiting times, and the variability of the waiting times.
### Step 1: List the Waiting Times
First, let's list the 19 waiting times:
[tex]\( 1, 3, 4, 5, 5, 6, 6, 7, 9, 9, 11, 12, 14, 15, 20, 22, 25, 30, 38 \)[/tex]
### Step 2: Calculate the Mean
The mean (average) waiting time is calculated by summing up all the waiting times and then dividing by the number of data points (19).
[tex]\[ \text{Mean} = \frac{\sum x}{n} \][/tex]
Where [tex]\( \sum x \)[/tex] is the sum of all waiting times, and [tex]\( n \)[/tex] is the number of waiting times.
[tex]\[ \sum x = 1 + 3 + 4 + 5 + 5 + 6 + 6 + 7 + 9 + 9 + 11 + 12 + 14 + 15 + 20 + 22 + 25 + 30 + 38 = 242 \][/tex]
[tex]\[ \text{Mean} = \frac{242}{19} \approx 12.736842105263158 \][/tex]
Thus, the mean waiting time is approximately [tex]\( 12.74 \)[/tex] minutes.
### Step 3: Calculate the Median
The median is the middle number in a sorted list of numbers. Since we have an odd number of waiting times (19), the median is the value at position [tex]\((n + 1) / 2 \)[/tex] in the ordered list.
The sorted waiting times are already given in order:
[tex]\[ 1, 3, 4, 5, 5, 6, 6, 7, 9, 9, 11, 12, 14, 15, 20, 22, 25, 30, 38 \][/tex]
The middle position is:
[tex]\((19 + 1) / 2 = 10 \)[/tex]
So, the median is the 10th number in the sorted list, which is [tex]\( 9 \)[/tex].
### Step 4: Calculate the Standard Deviation
The standard deviation measures the amount of variation or dispersion of a set of values. The formula for the standard deviation [tex]\( \sigma \)[/tex] is:
[tex]\[ \sigma = \sqrt{\frac{1}{n} \sum (x_i - \mu)^2} \][/tex]
Where:
- [tex]\( x_i \)[/tex] are the individual waiting times
- [tex]\( \mu \)[/tex] is the mean waiting time
- [tex]\( n \)[/tex] is the number of data points
We have already determined [tex]\( \mu \)[/tex] (the mean) to be approximately 12.74. After substituting the given waiting times and the calculated mean:
[tex]\[ \begin{aligned} \sigma &= \sqrt{\frac{1}{19} [(1-12.74)^2 + (3-12.74)^2 + \ldots + (38-12.74)^2]} \\ &\approx 9.775598542790416 \end{aligned} \][/tex]
Thus, the standard deviation is approximately [tex]\( 9.78 \)[/tex] minutes.
### Summary
- Mean: [tex]\( 12.74 \)[/tex] minutes
- Median: [tex]\( 9 \)[/tex] minutes
- Standard Deviation: [tex]\( 9.78 \)[/tex] minutes
These statistical measures provide valuable insights into the waiting times at Ellerton Bank, indicating the average time customers wait, the middle value of the waiting times, and the variability of the waiting times.
