Final answer:
To find the median daily wage of the data set, we calculated the cumulative frequencies and determined that the median falls within the 180-230 wage interval. Using the Midpoint Method, we estimated the median to be $205.
Step-by-step explanation:
To find the median of the given data, we first need to calculate the cumulative frequency of the number of labourers and then locate the middle value or values if the total number of observations is even. Here's a step-by-step explanation:
- Create a frequency table and include a column for the cumulative frequency.
- Find the total number of labourers by summing the frequencies.
- Determine the middle position(s) using the formula (N+1)/2 if there are an odd number of observations, or N/2 and (N/2) + 1 if there are even.
- Locate the median's position within the cumulative frequency to find the corresponding wage group.
- Since the wage groups are intervals, use the Midpoint Method to estimate the median within that interval.
Let's calculate the cumulative frequencies:
- 0-50: 10 labourers (cumulative frequency - 10)
- 60-110: 15 labourers (cumulative frequency - 25)
- 120-170: 12 labourers (cumulative frequency - 37)
- 180-230: 20 labourers (cumulative frequency - 57)
- 240-290: 13 labourers (cumulative frequency - 70)
Total number of labourers is 70. Since this is even, we take the average of the 35th and 36th values. Those values fall within the 180-230 wage interval.
We can estimate the median using the Midpoint Formula by taking the average of the lower and upper bounds of the 180-230 interval. The midpoint is therefore (180+230)/2 = 205.
The estimated median daily wage is $205.