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The table below shows the rates of births, marriages and deaths in Ireland from 1990 to 2010 - Leaving Cert Mathematics - Question Question 1 - 2012
Question Question 1
The table below shows the rates of births, marriages and deaths in Ireland from 1990 to 2010. The rates are per 10000 of the estimated population. | Year | Births |... show full transcript
Worked Solution & Example Answer:The table below shows the rates of births, marriages and deaths in Ireland from 1990 to 2010 - Leaving Cert Mathematics - Question Question 1 - 2012
Step 1
Complete the back to back stem and leaf plot.
Answer
To complete the back-to-back stem and leaf plot, the stem will represent the tens and the leaves the units of the marriage rates on one side and the death rates on the other.
- Marriage Rates: For example, for 49 it will be represented as:
- Stem: 4 | Leaf: 9
- Death Rates: For example, for 90 it will be:
- Stem: 8 | Leaf: 0 The completed plot shows a clearer visualization of the distributions.
Step 2
State one difference that can be observed between the distributions of the marriage rate and the death rate in your plot.
Answer
One noticeable difference is that the marriage rate appears to have a wider spread with higher values compared to the death rate, which seems to cluster more closely around the lower values, illustrating that marriage rates fluctuated more than death rates over the years.
Step 3
Find the median and interquartile range of the yearly marriage rates in Ireland from 1990 to 2010.
Answer
To find the median, arrange the marriage rates in ascending order: 49, 43, 43, 43, 45, 45, 51, 52, 52, 52, 54, 56, 56, 46. The median is the average of the 7th and 8th values, which is: Median = (51 + 52) / 2 = 51.5.
To find the interquartile range, determine the first (Q1) and third (Q3) quartiles:
- Q1 (25th percentile): 43
- Q3 (75th percentile): 56 Thus, the interquartile range is: IQR = Q3 - Q1 = 56 - 43 = 13.
Step 4
Find the mean of the death rate in Ireland from 1990 to 2010.
Answer
To calculate the mean death rate, sum all the death rates provided and divide by the number of years: Mean Death Rate = (90 + 89 + 88 + 87 + 90 + 90 + 87 + 86 + 87 + 87 + 83 + 81 + 71 + 67 + 67 + 64 + 64 + 66 + 63 + 66 + 61) / 21 = 78.3.
Step 5
List all of the death rates that are within 1 standard deviation of the mean.
Answer
The mean death rate calculated was 78.3 with a standard deviation of 10.3. Thus, we should look for rates between:
- Lower Bound: 78.3 - 10.3 = 68
- Upper Bound: 78.3 + 10.3 = 88.6
The death rates within this range (68 - 88) are:
- 71, 67, 67, 64, 64, 66, 66, 61.
Step 6
Estimate the total population of Ireland in 2010.
Answer
Given that the number of children born in Ireland in 2010 is 75,174 and represents the birth rate: Using the birth rate of 165 per 10,000, we can estimate the total population as: Total Population = (75,174 / 165) * 10,000 = 4,550,000.
Step 7
Step 8
Give a reason why the statement 'More children were born in Ireland in 1990 than in 2000' is not necessarily true.
Answer
While the raw number of births in 1990 (151) is higher than in 2000 (145), the significant difference in population size between those years must also be considered. Birth rates per capita may provide a different perspective.
Step 9
Find the ratio, Birth rate : Death rate, for the two years 1990 and 2010.
Answer
To find the ratios:
- For 1990: Birth Rate = 151, Death Rate = 90 → Ratio = 151:90 ≈ 1.68:1.
- For 2010: Birth Rate = 165, Death Rate = 61 → Ratio = 165:61 ≈ 2.70:1. These indicate a trend of increasing births relative to deaths over the years.
Step 10
Step 11
Describe the relationship between the birth rate and death rate.
Answer
The negative correlation coefficient of -0.85 indicates a strong inverse relationship. As the birth rate increases, the death rate tends to decrease. A potential reason could be improvements in healthcare and living conditions, leading to lower mortality rates.