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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
I would categorize the type of data collected as 'Numerical Continuous'.
Explanation: Height is a numerical measure that can take an infinite number of values within a given range, meaning it is measurable and can vary to any decimal point.
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 assign mid-interval values to each height range:
- 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
Next, we multiply each mid-value by the corresponding frequency and sum the products:
egin{align*} Mean Height, , ar{x} &= \frac{\sum (Mid-value \times Frequency)}{Total : Frequency} \ &= \frac{(147.5 \times 15) + (152.5 \times 48) + (157.5 \times 80) + (162.5 \times 112) + (167.5 \times 125) + (172.5 \times 81) + (177.5 \times 29) + (182.5 \times 10)}{500} \ &= \frac{82215}{500} \ &= 164.43 \text{ cm}
ight)
d= 164.43cm.
Step 3
Step 4
The median height of the girls in the sample is 164.5 cm. Explain what this means in the context of the heights of the 500 girls.
Answer
The median height of 164.5 cm indicates that half of the girls in this sample are shorter than 164.5 cm, while the other half are taller. This provides a central reference point for understanding the distribution of heights among the girls.
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 percentages, divide the number of girls in each height category by the total number of girls (500) and multiply by 100:
- 145-150 cm: (15/500) x 100 = 3.0%
- 150-155 cm: (48/500) x 100 = 9.6%
- 155-160 cm: (80/500) x 100 = 16.0%
- 160-165 cm: (112/500) x 100 = 22.4%
- 165-170 cm: (125/500) x 100 = 25.0%
- 170-175 cm: (81/500) x 100 = 16.2%
- 175-180 cm: (29/500) x 100 = 5.8%
- 180-185 cm: (10/500) x 100 = 2.0%
Thus, we would complete Table 2.
Step 6
Use the data in Table 2 to draw a histogram showing the percentage of girls in each height category.
Answer
To draw the histogram:
- Label the x-axis with the height categories and the y-axis with percentage values.
- Plot each percentage over the corresponding height category using bars, ensuring the heights are visually represented correctly.
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 boys in the 175-180 cm category is approximately twice that of the area for girls in the same height category, indicating a greater percentage of boys.
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 her claim. The combined areas of the height categories from 165 cm to 185 cm in the boys’ histogram exceed that of the girls’, showing more boys than tall girls.
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:
- Mean (ar{x}) = 166.7 cm
- Standard Deviation () = 8.9 cm
Approximate interval for 95% of boys' heights:
egin{align*} ext{Interval} &= \bar{x} \pm 2\text{s} \ &= 166.7 \pm 2(8.9) \ &= [166.7 - 17.8, 166.7 + 17.8] \ &= [148.9, 184.5]
ight)
d= [148.9, 184.5] cm.
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 smaller standard deviation for girls (7.7 cm) compared to boys (8.9 cm) suggests that the heights of the girls are more closely clustered around their mean than those of the boys, indicating less variability in girls' heights.