### **Statistics Exercise (Data Handling & Interpretation)**
**Instructions:** Answer all questions. Show calculations where necessary.
---
### **Section A: Stem-and-Leaf Diagrams**
**Question 1:**
The stem-and-leaf plot shows the weights (in kg) of 15 students:
```
Stem | Leaf
4 |258
5 |01367
6 |2459
7 |13
```
a) What is the range of the data?
b) Find the median weight.
c) Calculate the mean weight (to 1 decimal place).
d) How many students weigh more than 55 kg?
**Answers:**
a) \(72 - 42 = 30\) kg
b) Median = \(56\) kg
c) Mean = \(\frac{42+45+48+50+51+53+56+57+62+64+65+69+71+73}{15} = 57.5\) kg
d) **9 students** (56, 57, 62, 64, 65, 69, 71, 73)
---
### **Section B: Pie Charts**
**Question 2:**
A pie chart shows the favorite fruits of 120 people:
- Apple: 90°
- Banana: 60°
- Orange: 120°
- Grapes: 90°
a) How many people prefer apples?
b) What percentage like bananas?
c) If 15 more people choose grapes, what is the new angle?
d) Draw the pie chart with the given data.
**Answers:**
a) \(\frac{90°}{360°} \times 120 = 30\) people
b) \(\frac{60°}{360°} \times 100 = 16.67\%\)
c) New grapes voters = 30 + 15 = 45 → Angle = \(\frac{45}{120} \times 360° = 135°\)
d) *(Draw accordingly)*
---
### **Section C: Frequency Tables**
**Question 3:**
The table shows marks scored by students in a test:
| Marks | Frequency |
|-------|-----------|
| 0-10 | 4 |
| 10-20 | 8 |
| 20-30 | 12 |
| 30-40 | 6 |
a) How many students took the test?
b) What is the modal class?
c) Estimate the mean mark.
d) Draw a histogram for the data.
**Answers:**
a) \(4 + 8 + 12 + 6 = 30\) students
b) **20-30** (highest frequency)
c) Mean ≈ \(\frac{(5×4)+(15×8)+(25×12)+(35×6)}{30} = 21\)
d) *(Draw with appropriate bars)*
---
### **Section D: Cumulative Frequency & Box Plots**
**Question 4:**
The cumulative frequency table shows the time taken (in minutes) by 50 workers to complete a
task:
| Time (min) | Cumulative Freq |
|------------|-----------------|
| 10-20 |5 |
| 20-30 | 18 |
| 30-40 | 34 |
| 40-50 | 50 |
a) How many workers took 25-35 minutes?
b) Find the median time.
c) Calculate the interquartile range (IQR).
d) Draw a box plot for the data.
**Answers:**
a) \(34 - 5 = 29\) workers
b) Median = 30-40 class → Estimate ≈ 32.5 min
c) Q1 = 20-30 class (~23 min), Q3 = 30-40 class (~37 min) → IQR = 14 min
d) *(Plot with min=10, Q1=23, median=32.5, Q3=37, max=50)*
---
### **Section E: Probability & Two-Way Tables**
**Question 5:**
A survey of 100 people on their preference for tea or coffee:
| | Tea | Coffee | Total |
|-------|-----|--------|-------|
| Men | 30 | 20 | 50 |
| Women | 25 | 25 | 50 |
| Total | 55 | 45 | 100 |
a) What is P(Men who prefer tea)?
b) Find P(Coffee | Women).
c) Are gender and drink preference independent? Justify.
d) If 2 people are randomly picked, what is P(Both prefer tea)?
**Answers:**
a) \(\frac{30}{100} = 0.3\)
b) \(\frac{25}{50} = 0.5\)
c) **No**, because P(Coffee) = 0.45 ≠ P(Coffee | Women) = 0.5.
d) \(\frac{55}{100} \times \frac{54}{99} = 0.297\)
---
### **Section F: Scatter Graphs & Correlation**
**Question 6:**
The table shows hours studied (x) and test scores (y):
| Hours (x) | 2 | 4 | 5 | 7 | 9 |
| Score (y) | 30| 50| 60| 80| 90|
a) Plot the scatter graph.
b) Describe the correlation.
c) Find the equation of the line of best fit (use \(y = mx + c\)).
d) Predict the score for 6 hours of study.
**Answers:**
b) **Strong positive correlation**
c) Mean \(x = 5.4\), \(y = 62\) → \(y ≈ 10x + 8\) *(approximate)*
d) \(y ≈ 10(6) + 8 = 68\)
---
### **Section G: Histograms & Frequency Density**
**Question 7:**
The table shows the ages of attendees at a concert:
| Age (years) | Frequency |
|-------------|-----------|
| 10-15 | 20 |
| 15-20 | 30 |
| 20-30 | 40 |
| 30-50 | 10 |
a) Why is a histogram suitable?
b) Calculate frequency density for each class.
c) Draw the histogram.
d) Estimate the number of people aged 12-18.
**Answers:**
a) **Unequal class widths**
b) FD = Freq/Class Width → 4, 6, 4, 0.5
d) \(\frac{3}{5} \times 20 + 30 = 42\)
---
### **Section H: Time Series & Moving Averages**
**Question 8:**
The data shows monthly sales (in $1000s):
| Month | Jan | Feb | Mar | Apr | May |
| Sales | 12 | 15 | 18 | 14 | 16 |
a) Plot the time series graph.
b) Calculate the 3-month moving averages.
c) Predict June sales using the trend.
d) Comment on seasonality.
**Answers:**
b) Feb: \(\frac{12+15+18}{3} = 15\), Mar: \(\frac{15+18+14}{3} ≈ 15.67\)
c) **≈17** (upward trend)
d) **No clear seasonality** (insufficient data)
---
### **Section I: Probability Distributions**
**Question 9:**
A fair die is rolled. The probability distribution of X (outcome) is:
|X|1|2|3|4|5|6|
| P(X) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) |
a) Find E(X).
b) Calculate Var(X).
c) If Y = 2X + 1, find E(Y).
d) What is P(X > 4)?
**Answers:**
a) \(E(X) = 3.5\)
b) \(Var(X) = \frac{35}{12} ≈ 2.92\)
c) \(E(Y) = 2(3.5) + 1 = 8\)
d) \(\frac{2}{6} = \frac{1}{3}\)
---
### **Section J: Real-World Applications**
**Question 10:**
A school measures heights (cm) of 50 students:
| Height (cm) | Freq |
|-------------|------|
| 140-150 |5 |
| 150-160 | 12 |
| 160-170 | 18 |
| 170-180 | 10 |
| 180-190 |5 |
a) Find the median height.
b) Calculate the standard deviation (assume mean = 165 cm).
c) Draw a cumulative frequency curve.
d) Estimate how many students are taller than 175 cm.
**Answers:**
a) Median ≈ 163.3 cm (from CF curve)
b) Assume σ ≈ 10 cm *(approximate)*
d) **≈8 students**
---
**End of Exercise**
😊
This exercise covers **stem-and-leaf plots, pie charts, frequency tables, probability, correlation,
histograms, and real-world data analysis**. Let me know if you need adjustments!
