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Some students are using a database of earthquakes to investigate the times between the occurrences of serious earthquakes around the world - Leaving Cert Mathematics - Question 7 - 2011
Question 7
Some students are using a database of earthquakes to investigate the times between the occurrences of serious earthquakes around the world. They extract information ... show full transcript
Worked Solution & Example Answer:Some students are using a database of earthquakes to investigate the times between the occurrences of serious earthquakes around the world - Leaving Cert Mathematics - Question 7 - 2011
Step 1
Create a suitable graphical representation of the distribution.
Answer
To effectively visualize the distribution of earthquakes based on the time intervals, a histogram is created. The x-axis represents the time in days from the previous earthquake, divided into intervals, while the y-axis indicates the number of earthquakes in each interval. For this data:
- The intervals include: 0-100, 100-200, 200-300, etc.
- The height of each bar corresponds to the number of earthquakes that fall within each time interval.
This method provides a clear depiction of how earthquakes are distributed over time intervals.
Step 2
Describe the distribution.
Answer
The distribution is observed to be positively skewed, meaning that it has a tail extending towards the right. This implies that there are fewer time intervals with very long gaps between earthquakes compared to those with shorter intervals. Moreover, the median of the distribution is estimated at approximately 220 days, which indicates that half of the observed time intervals fall below this value.
Step 3
Explain why it would not be correct to use standard normal distribution tables (z-tables) to do this.
Answer
Using standard normal distribution tables is inappropriate in this case because the distribution of earthquake intervals is not normal; it is positively skewed. The mean and the median do not align in skewed distributions, and thus normal probabilities derived from z-tables would not accurately reflect the real probabilities of the time until the next earthquake.
Step 4
What is the best estimate for the probability described in part (iii) above?
Answer
The best estimate of the probability for the intervals between 100 and 200 days is given by the proportion of historical data that falls within this range. Given the data:
ext{Probability} = rac{24}{115}
This fraction reflects the estimate based on historical occurrences and provides a direct correlation to past data.
Step 5
Suggest a different way that they could have looked at the data.
Answer
Instead of just analyzing the time intervals, the students could have also examined the data by looking at:
- The number of earthquakes per decade, which may reveal trends over time.
- Categorizing earthquakes based on their magnitudes and their subsequent impacts.
- Taking into account geographical regions and their seismic activity.
These approaches may uncover additional patterns and facilitate better understanding of earthquake occurrences.