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Show the annual total sales of cars over the six years, using a suitable chart - Leaving Cert Mathematics - Question 1 - 2012
Question 1
Show the annual total sales of cars over the six years, using a suitable chart. Find the mean number of cars sold per year over the six years. Calculate the percen... show full transcript
Worked Solution & Example Answer:Show the annual total sales of cars over the six years, using a suitable chart - Leaving Cert Mathematics - Question 1 - 2012
Step 1
Show the annual total sales of cars over the six years, using a suitable chart.
Answer
To represent the annual total sales of cars over the six years, a bar chart is appropriate. The x-axis should indicate the years (2006 to 2011) and the y-axis should indicate the annual total sales in thousands. Each bar should correspond to the total sales for each year, visually demonstrating trends or fluctuations.
Step 2
Find the mean number of cars sold per year over the six years.
Answer
To calculate the mean number of cars sold per year, sum all the cars sold from 2006 to 2011 and divide by the number of years. The formula is:
ext{Mean} = rac{ ext{Total Sales}}{n}
where Total Sales = 173273 + 180754 + 146470 + 544432 + 84907 + 86932 = 726768 and n = 6. Thus,
ext{Mean} = rac{726768}{6} = 121128
Step 3
Calculate the percentage increase in car sales between 2009 and 2011.
Answer
The percentage increase in car sales from 2009 to 2011 can be calculated using the formula:
ext{Percentage Increase} = rac{ ext{New Value} - ext{Old Value}}{ ext{Old Value}} imes 100
Here, the New Value (2011 sales) = 86932 and the Old Value (2009 sales) = 54432. Hence,
ext{Percentage Increase} = rac{86932 - 54432}{54432} imes 100 = rac{32500}{54432} imes 100 ext{ which is approximately } 59.71 ext{%}
Step 4
Give a more balanced description of the pattern of car sales over the six years.
Answer
The pattern of car sales shows a dramatic decline from 2007 to 2009, indicating a challenging market environment. However, post-2009, sales exhibit signs of recovery, although they remain significantly lower than the peaks recorded in 2007. Notably, while 2010 and 2011 show improved figures, they still do not reach the highs of previous years.
Step 5
Describe how the sales of the cars are distributed over the four quarters of each year.
Answer
Sales are typically highest in the first quarter of each year, likely due to consumer behavior related to new year registrations. There is a gradual decline in sales in the subsequent quarters, indicating a seasonal pattern where consumers may be less inclined to purchase new cars after the initial quarter.
Step 6
Suggest a reason for this pattern of sales.
Answer
One primary reason for the sales pattern is that many consumers prefer to buy new cars at the beginning of the year to obtain the latest models, often associated with new registration plates in many regions. This trend results in a spike in sales during the first quarter.
Step 7
Find, with justification, an estimate for the total annual sales for 2012.
Answer
The first quarter sales for 2012 are given as 36,081. Assuming that this represents about one-third of the total annual sales based on previous years' performance:
Total estimate for 2012 can be calculated as:
Rounding this could provide an estimate in the region of 85,500 based on a similar proportion observed in 2010 and 2011.
Step 8
Construct a back-to-back stem-and-leaf plot of the above data.
Answer
The stem-and-leaf plot should be constructed with stems representing the tens place and leaves representing the units place of the given data. An example format is:
Back-to-back stem-and-leaf plot:
Stem | Diesel | Petrol
-------|--------|-------
7 | 9,8,8,7,6,4,0 | 11
5 | 5,6,0 | 12
4 | 3,4,6,8,9 | 14
15 | 0,7,9
Key: 12 | 9 means 129
This construction highlights the distribution of diesel and petrol emissions effectively.
Step 9
Does the information suggest that diesel engines produce lower CO2 emissions than petrol engines?
Answer
Analyzing the stem-and-leaf plot, it appears that the diesel engines are grouped more closely at lower values, indicating a smaller median. This suggests that, on average, diesel engines may produce lower CO2 emissions compared to petrol engines.
Step 10
Does the information suggest that there is greater variation in the CO2 emissions of diesel engines than petrol engines?
Answer
From the stem-and-leaf plot, the data for petrol engines is spread over a wider range as the leaves indicate more variability. This suggests that the CO2 emissions from petrol engines exhibit greater variation compared to those from diesel engines, which are more clustered around lower values.