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The heights of a random sample of 1000 students were collected and recorded - Leaving Cert Mathematics - Question 9 - 2015
Question 9
The heights of a random sample of 1000 students were collected and recorded. (a) Tick one box from the table below to indicate how you would categorise the type of ... show full transcript
Worked Solution & Example Answer:The heights of a random sample of 1000 students were collected and recorded - Leaving Cert Mathematics - Question 9 - 2015
Step 1
Tick one box from the table below to indicate how you would categorise the type of data collected. Explain your choice.
Answer
The type of data collected is 'Numerical Continuous' because height is measured on a continuous scale and can take any value within a certain range. This means that heights can be decimals and are not limited to whole numbers.
Step 2
Use the information in Table 1 to estimate the mean height of the girls, using mid-interval values.
Answer
To find the mean height, we will calculate the midpoints for each height category and then determine the weighted average. The midpoints are:
- 145-150 cm: 147.5 cm
- 150-155 cm: 152.5 cm
- 155-160 cm: 157.5 cm
- 160-165 cm: 162.5 cm
- 165-170 cm: 167.5 cm
- 170-175 cm: 172.5 cm
- 175-180 cm: 177.5 cm
- 180-185 cm: 182.5 cm
Now calculating the mean:
Step 3
What is the largest possible value for the range of the heights of the girls in this sample?
Answer
The largest possible value for the range can be calculated by subtracting the lowest height from the highest height. The lowest category is 145 cm and the highest is 185 cm. Thus, the range is:
Step 4
The median height of the girls in the sample is 164.5 cm. Explain what this means in the context of the 500 girls.
Answer
The median height of 164.5 cm indicates that half of the 500 girls are shorter than or equal to 164.5 cm and the other half are taller than or equal to this height. This provides a useful measure of the central tendency of the girls' heights.
Step 5
Use the data in Table 1 to complete Table 2 by finding the percentage of girls in each of the height categories.
Answer
To find the percentage, we calculate:
- For 145-150 cm: ( rac{15}{500} \times 100 = 3.0%)
- For 150-155 cm: ( rac{48}{500} \times 100 = 9.6%)
- For 155-160 cm: ( rac{80}{500} \times 100 = 16.0%)
- For 160-165 cm: ( rac{112}{500} \times 100 = 22.4%)
- For 165-170 cm: ( rac{125}{500} \times 100 = 25.0%)
- For 170-175 cm: ( rac{81}{500} \times 100 = 16.2%)
- For 175-180 cm: ( rac{29}{500} \times 100 = 5.8%)
- For 180-185 cm: ( rac{10}{500} \times 100 = 2.0%)
Step 6
Use the data in Table 2 to draw a histogram showing the percentage of girls in each height category.
Answer
The histogram would have height categories on the x-axis and percentage of girls on the y-axis. Each bar would correspond to the height ranges with the appropriate percentages calculated previously.
Step 7
A histogram showing the percentage of boys in each height category is given above. John examines both histograms and comments that "There are roughly twice as many boys as girls in the 175 to 180 cm category". Do the histograms support his claim? Explain your answer.
Answer
Yes, the histograms support John's claim. The area of the bar for the category 175-180 in the boys' histogram is approximately twice the area of the corresponding category in the girls' histogram.
Step 8
Many examines both histograms and comments that "I see that there are more tall girls than tall boys". Do the two histograms support her claim? Explain your answer.
Answer
No, the histograms do not support this claim. The combined areas of the height categories from 165 cm onwards show a greater number of boys compared to the number of girls, indicating that there are more boys than girls taller than 165 cm.
Step 9
The mean height of the boys in the sample is 166.7 cm and the standard deviation of their height is 8.9 cm. Assuming that boys' heights are normally distributed, use the Empirical Rule to find an interval that will contain the heights of approximately 95% of all boys.
Answer
Using the Empirical Rule, approximately 95% of the boys' heights will fall within two standard deviations of the mean. Therefore, we calculate:
Step 10
The standard deviation of the heights of the girls in the sample is 7.7 cm while the standard deviation of the heights of the boys is 8.9 cm. Interpret this difference in the context of the data.
Answer
The standard deviation indicates the variability of height within each group. A smaller standard deviation for girls (7.7 cm) suggests that the heights of the girls are more clustered around the mean compared to boys, who have a greater variability in their heights (8.9 cm). This indicates that there is more diversity in boys' heights than in girls' heights.