Answer :
To solve the given questions, we need to calculate various statistical measures for the given list of ticket prices:
[tex]\[ \$75, 120, 120, 145, 150, 150, 150, 175, 175, 200, 225, 275 \][/tex]
Here are the steps and detailed calculations for each part of the problem:
### Part (a): Calculate the Mean Ticket Price
The mean is the average of all the ticket prices. It is calculated by summing all the values and dividing by the number of values.
[tex]\[
\text{Mean} = \frac{\sum \text{prices}}{n}
\][/tex]
Sum of the ticket prices:
[tex]\[
75 + 120 + 120 + 145 + 150 + 150 + 150 + 175 + 175 + 200 + 225 + 275 = 1960
\][/tex]
Number of tickets:
[tex]\[
n = 12
\][/tex]
Mean:
[tex]\[
\text{Mean} = \frac{1960}{12} \approx 163.33
\][/tex]
So, the mean ticket price is [tex]\(\$163.33\)[/tex].
### Part (b): Calculate the Median Ticket Price
The median is the middle value in a list when the values are arranged in ascending order. If the number of observations is even, the median is the average of the two middle numbers.
Arranging the prices in ascending order:
[tex]\[
75, 120, 120, 145, 150, 150, 150, 175, 175, 200, 225, 275
\][/tex]
There are 12 values, so the median is the average of the 6th and 7th values:
[tex]\[
\text{Median} = \frac{150 + 150}{2} = 150
\][/tex]
So, the median ticket price is [tex]\(\$150.00\)[/tex].
### Part (c): Calculate the Mode Ticket Price
The mode is the value that appears most frequently in the list.
In this list, the value 150 appears 3 times, which is more than any other value.
So, the mode ticket price is [tex]\(\$150.00\)[/tex].
### Part (d): Calculate the Range
The range is the difference between the maximum and minimum values.
[tex]\[
\text{Range} = \text{Maximum price} - \text{Minimum price}
\][/tex]
Maximum price:
[tex]\[
275
\][/tex]
Minimum price:
[tex]\[
75
\][/tex]
Range:
[tex]\[
\text{Range} = 275 - 75 = 200
\][/tex]
So, the range is [tex]\(\$200.00\)[/tex].
### Part (e): Calculate the Variance
Variance measures how spread out the values are around the mean. For the sample variance, we use the formula:
[tex]\[
\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1}
\][/tex]
where [tex]\( x_i \)[/tex] is each individual value, [tex]\( \bar{x} \)[/tex] is the mean, and [tex]\( n \)[/tex] is the number of observations.
First, we calculate the squared differences from the mean for each value, then sum these squared differences:
[tex]\[
\sum (x_i - \bar{x})^2 = (75 - 163.33)^2 + (120 - 163.33)^2 + \ldots + (275 - 163.33)^2
\][/tex]
Calculating each squared difference:
[tex]\[
(75 - 163.33)^2 = 7834.22
\][/tex]
[tex]\[
(120 - 163.33)^2 = 1870.00
\][/tex]
[tex]\[
(120 - 163.33)^2 = 1870.00
\][/tex]
[tex]\[
(145 - 163.33)^2 = 336.10
\][/tex]
[tex]\[
(150 - 163.33)^2 = 177.68
\][/tex]
[tex]\[
(150 - 163.33)^2 = 177.68
\][/tex]
[tex]\[
(150 - 163.33)^2 = 177.68
\][/tex]
[tex]\[
(175 - 163.33)^2 = 136.14
\][/tex]
[tex]\[
(175 - 163.33)^2 = 136.14
\][/tex]
[tex]\[
(200 - 163.33)^2 = 1335.62
\][/tex]
[tex]\[
(225 - 163.33)^2 = 3797.78
\][/tex]
[tex]\[
(275 - 163.33)^2 = 12489.11
\][/tex]
Summing these:
[tex]\[
7834.22 + 1870.00 + 1870.00 + 336.10 + 177.68 + 177.68 + 177.68 + 136.14 + 136.14 + 1335.62 + 3797.78 + 12489.11 = 29338.15
\][/tex]
Variance:
[tex]\[
\text{Variance} = \frac{29338.15}{11} \approx 2667.10
\][/tex]
So, the variance is [tex]\(2667.10\)[/tex].
### Part (f): Calculate the Standard Deviation
The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\][/tex]
[tex]\[
\text{Standard Deviation} = \sqrt{2667.10} \approx 51.65
\][/tex]
So, the standard deviation is [tex]\(51.65\)[/tex].
### Summary of Answers
a. Mean ticket price: [tex]\(\$163.33\)[/tex]
b. Median ticket price: [tex]\(\$150.00\)[/tex]
c. Mode ticket price: [tex]\(\$150.00\)[/tex]
d. Range: [tex]\(\$200.00\)[/tex]
e. Variance: [tex]\(2667.10\)[/tex]
f. Standard Deviation: [tex]\(51.65\)[/tex]
[tex]\[ \$75, 120, 120, 145, 150, 150, 150, 175, 175, 200, 225, 275 \][/tex]
Here are the steps and detailed calculations for each part of the problem:
### Part (a): Calculate the Mean Ticket Price
The mean is the average of all the ticket prices. It is calculated by summing all the values and dividing by the number of values.
[tex]\[
\text{Mean} = \frac{\sum \text{prices}}{n}
\][/tex]
Sum of the ticket prices:
[tex]\[
75 + 120 + 120 + 145 + 150 + 150 + 150 + 175 + 175 + 200 + 225 + 275 = 1960
\][/tex]
Number of tickets:
[tex]\[
n = 12
\][/tex]
Mean:
[tex]\[
\text{Mean} = \frac{1960}{12} \approx 163.33
\][/tex]
So, the mean ticket price is [tex]\(\$163.33\)[/tex].
### Part (b): Calculate the Median Ticket Price
The median is the middle value in a list when the values are arranged in ascending order. If the number of observations is even, the median is the average of the two middle numbers.
Arranging the prices in ascending order:
[tex]\[
75, 120, 120, 145, 150, 150, 150, 175, 175, 200, 225, 275
\][/tex]
There are 12 values, so the median is the average of the 6th and 7th values:
[tex]\[
\text{Median} = \frac{150 + 150}{2} = 150
\][/tex]
So, the median ticket price is [tex]\(\$150.00\)[/tex].
### Part (c): Calculate the Mode Ticket Price
The mode is the value that appears most frequently in the list.
In this list, the value 150 appears 3 times, which is more than any other value.
So, the mode ticket price is [tex]\(\$150.00\)[/tex].
### Part (d): Calculate the Range
The range is the difference between the maximum and minimum values.
[tex]\[
\text{Range} = \text{Maximum price} - \text{Minimum price}
\][/tex]
Maximum price:
[tex]\[
275
\][/tex]
Minimum price:
[tex]\[
75
\][/tex]
Range:
[tex]\[
\text{Range} = 275 - 75 = 200
\][/tex]
So, the range is [tex]\(\$200.00\)[/tex].
### Part (e): Calculate the Variance
Variance measures how spread out the values are around the mean. For the sample variance, we use the formula:
[tex]\[
\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1}
\][/tex]
where [tex]\( x_i \)[/tex] is each individual value, [tex]\( \bar{x} \)[/tex] is the mean, and [tex]\( n \)[/tex] is the number of observations.
First, we calculate the squared differences from the mean for each value, then sum these squared differences:
[tex]\[
\sum (x_i - \bar{x})^2 = (75 - 163.33)^2 + (120 - 163.33)^2 + \ldots + (275 - 163.33)^2
\][/tex]
Calculating each squared difference:
[tex]\[
(75 - 163.33)^2 = 7834.22
\][/tex]
[tex]\[
(120 - 163.33)^2 = 1870.00
\][/tex]
[tex]\[
(120 - 163.33)^2 = 1870.00
\][/tex]
[tex]\[
(145 - 163.33)^2 = 336.10
\][/tex]
[tex]\[
(150 - 163.33)^2 = 177.68
\][/tex]
[tex]\[
(150 - 163.33)^2 = 177.68
\][/tex]
[tex]\[
(150 - 163.33)^2 = 177.68
\][/tex]
[tex]\[
(175 - 163.33)^2 = 136.14
\][/tex]
[tex]\[
(175 - 163.33)^2 = 136.14
\][/tex]
[tex]\[
(200 - 163.33)^2 = 1335.62
\][/tex]
[tex]\[
(225 - 163.33)^2 = 3797.78
\][/tex]
[tex]\[
(275 - 163.33)^2 = 12489.11
\][/tex]
Summing these:
[tex]\[
7834.22 + 1870.00 + 1870.00 + 336.10 + 177.68 + 177.68 + 177.68 + 136.14 + 136.14 + 1335.62 + 3797.78 + 12489.11 = 29338.15
\][/tex]
Variance:
[tex]\[
\text{Variance} = \frac{29338.15}{11} \approx 2667.10
\][/tex]
So, the variance is [tex]\(2667.10\)[/tex].
### Part (f): Calculate the Standard Deviation
The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\][/tex]
[tex]\[
\text{Standard Deviation} = \sqrt{2667.10} \approx 51.65
\][/tex]
So, the standard deviation is [tex]\(51.65\)[/tex].
### Summary of Answers
a. Mean ticket price: [tex]\(\$163.33\)[/tex]
b. Median ticket price: [tex]\(\$150.00\)[/tex]
c. Mode ticket price: [tex]\(\$150.00\)[/tex]
d. Range: [tex]\(\$200.00\)[/tex]
e. Variance: [tex]\(2667.10\)[/tex]
f. Standard Deviation: [tex]\(51.65\)[/tex]
