Answer :
To solve part (a) of the problem, we need to identify the outliers in the given dataset. Let's follow a step-by-step approach.
### Step 1: Organize the Data
Given data:
[tex]\[ 0, 282, 158, 408, 0, 98, 287, 104, 403, 116, 3043, 589, 547, 0, 0, 150, 0, 281, 172, 275 \][/tex]
### Step 2: Calculate the Quartiles
First, arrange the data in ascending order:
[tex]\[ 0, 0, 0, 0, 0, 98, 104, 116, 150, 158, 172, 275, 281, 282, 287, 403, 408, 547, 589, 3043 \][/tex]
Q1 (25th percentile): The median of the first half of the data (before the median of the entire data set).
[tex]\[ Q1 = \frac{0 + 104}{2} = 52 \][/tex]
Q3 (75th percentile): The median of the second half of the data (after the median of the entire data set).
[tex]\[ Q3 = \frac{287 + 403}{2} = 293 \][/tex]
### Step 3: Calculate the Interquartile Range (IQR)
[tex]\[ IQR = Q3 - Q1 = 293 - 52 = 241 \][/tex]
### Step 4: Determine the Outlier Cutoffs
Outliers are typically defined as values below [tex]\( Q1 - 1.5 \times IQR \)[/tex] or above [tex]\( Q3 + 1.5 \times IQR \)[/tex].
Lower bound:
[tex]\[ \text{Lower bound} = Q1 - 1.5 \times IQR = 52 - 1.5 \times 241 = 52 - 361.5 = -309.5 \][/tex]
Upper bound:
[tex]\[ \text{Upper bound} = Q3 + 1.5 \times IQR = 293 + 1.5 \times 241 = 293 + 361.5 = 654.5 \][/tex]
### Step 5: Identify the Outliers
Any value below -309.5 or above 654.5 is considered an outlier in the data set.
Reviewing the data:
[tex]\[ 0, 0, 0, 0, 0, 98, 104, 116, 150, 158, 172, 275, 281, 282, 287, 403, 408, 547, 589, 3043 \][/tex]
We see that only one value exceeds 654.5:
[tex]\[ 3043 \][/tex]
Hence, the outlier is:
[tex]\[ 3043 \][/tex]
### Answer for Part (a)
The outlier(s) is/are [tex]\( 3043 \)[/tex].
### Part (b): Drawing a Histogram
To draw a histogram, we need to group the income data into intervals (also called bins). The width of each bin can be chosen such that it reasonably represents the data. Let's choose appropriate bins and draw the histogram:
1. 0-100
2. 101-200
3. 201-300
4. 301-400
5. 401-500
6. 501-600
7. 601 and above
Next, tally the number of data points in each interval and plot the histogram. You'll see most of the student incomes are clustered towards the lower end with one high outlier (3043). (Since I can't physically plot the histogram here, you would need to use a graphing tool or graph paper to do so.)
### Part (c): Explanation for Any Outliers
Outliers are values that are significantly higher or lower than the rest of the data. They can occur due to various reasons:
- Data entry errors.
- Unusual but genuine variations.
- Specific subsets within the general population with different characteristics.
In this data set, the income of 3043 is much higher than that of the other students. This could be due to a unique situation, such as:
- The student might have a high-paying job or multiple sources of income.
- This income might represent an anomaly, such as a one-time payment, bonus, or an error in recording the income.
In summary:
- (a) The outlier(s) is/are [tex]\( 3043 \)[/tex].
- (b) A histogram should be plotted on graph paper or using a graphing tool with the indicated bins.
- (c) The outlier is likely due to an unusually high income for a student, which could be due to various reasons like a high-paying job or an unusual payment that week.
### Step 1: Organize the Data
Given data:
[tex]\[ 0, 282, 158, 408, 0, 98, 287, 104, 403, 116, 3043, 589, 547, 0, 0, 150, 0, 281, 172, 275 \][/tex]
### Step 2: Calculate the Quartiles
First, arrange the data in ascending order:
[tex]\[ 0, 0, 0, 0, 0, 98, 104, 116, 150, 158, 172, 275, 281, 282, 287, 403, 408, 547, 589, 3043 \][/tex]
Q1 (25th percentile): The median of the first half of the data (before the median of the entire data set).
[tex]\[ Q1 = \frac{0 + 104}{2} = 52 \][/tex]
Q3 (75th percentile): The median of the second half of the data (after the median of the entire data set).
[tex]\[ Q3 = \frac{287 + 403}{2} = 293 \][/tex]
### Step 3: Calculate the Interquartile Range (IQR)
[tex]\[ IQR = Q3 - Q1 = 293 - 52 = 241 \][/tex]
### Step 4: Determine the Outlier Cutoffs
Outliers are typically defined as values below [tex]\( Q1 - 1.5 \times IQR \)[/tex] or above [tex]\( Q3 + 1.5 \times IQR \)[/tex].
Lower bound:
[tex]\[ \text{Lower bound} = Q1 - 1.5 \times IQR = 52 - 1.5 \times 241 = 52 - 361.5 = -309.5 \][/tex]
Upper bound:
[tex]\[ \text{Upper bound} = Q3 + 1.5 \times IQR = 293 + 1.5 \times 241 = 293 + 361.5 = 654.5 \][/tex]
### Step 5: Identify the Outliers
Any value below -309.5 or above 654.5 is considered an outlier in the data set.
Reviewing the data:
[tex]\[ 0, 0, 0, 0, 0, 98, 104, 116, 150, 158, 172, 275, 281, 282, 287, 403, 408, 547, 589, 3043 \][/tex]
We see that only one value exceeds 654.5:
[tex]\[ 3043 \][/tex]
Hence, the outlier is:
[tex]\[ 3043 \][/tex]
### Answer for Part (a)
The outlier(s) is/are [tex]\( 3043 \)[/tex].
### Part (b): Drawing a Histogram
To draw a histogram, we need to group the income data into intervals (also called bins). The width of each bin can be chosen such that it reasonably represents the data. Let's choose appropriate bins and draw the histogram:
1. 0-100
2. 101-200
3. 201-300
4. 301-400
5. 401-500
6. 501-600
7. 601 and above
Next, tally the number of data points in each interval and plot the histogram. You'll see most of the student incomes are clustered towards the lower end with one high outlier (3043). (Since I can't physically plot the histogram here, you would need to use a graphing tool or graph paper to do so.)
### Part (c): Explanation for Any Outliers
Outliers are values that are significantly higher or lower than the rest of the data. They can occur due to various reasons:
- Data entry errors.
- Unusual but genuine variations.
- Specific subsets within the general population with different characteristics.
In this data set, the income of 3043 is much higher than that of the other students. This could be due to a unique situation, such as:
- The student might have a high-paying job or multiple sources of income.
- This income might represent an anomaly, such as a one-time payment, bonus, or an error in recording the income.
In summary:
- (a) The outlier(s) is/are [tex]\( 3043 \)[/tex].
- (b) A histogram should be plotted on graph paper or using a graphing tool with the indicated bins.
- (c) The outlier is likely due to an unusually high income for a student, which could be due to various reasons like a high-paying job or an unusual payment that week.
