To address your questions systematically, I’ll calculate each requested value step by step. We'll use the given data:

Data (in months): 5, 8, 10, 11, 11, 14, 15, 17, 23, 26, 26, 28, 29, 30, 31, 32, 34, 39, 39, 45

Step-by-Step Calculations:

(c) Calculate the value (Quartile 1, Q1):

1. Arrange the data in ascending order (already done).

2. Locate Q1 using the formula: $Q_1 = \text{Value at position } \frac{n+1}{4}$, where $n = 20$ (number of data points).

   $Q_1 = \text{Value at position } \frac{20+1}{4} = \text{Value at position } 5.25$

   Since 5.25 is between 5th and 6th positions, interpolate:

   • 5th value = 11
   • 6th value = 14
   • Interpolation: $11 + 0.25 \times (14 - 11) = 11 + 0.75 = 11.75$

   Q1 = 11.75

(d) Calculate the value (Median/Q2):

1. Median is the value at position $\frac{n+1}{2}$: $Q_2 = \text{Value at position } \frac{20+1}{2} = \text{Value at position } 10.5$

   Interpolate:

   • 10th value = 26
   • 11th value = 26
   • Median: $26 + 0.5 \times (26 - 26) = 26$

   Q2 (Median) = 26

(e) Calculate the value (Quartile 3, Q3):

1. Locate Q3 using the formula: $Q_3 = \text{Value at position } 3 \times \frac{n+1}{4} = 3 \times \frac{20+1}{4} = \text{Value at position } 15.75$

   Interpolate:

   • 15th value = 31
   • 16th value = 32
   • Interpolation: $31 + 0.75 \times (32 - 31) = 31 + 0.75 = 31.75$

   Q3 = 31.75

(f) Interquartile Range (IQR):

$IQR = Q_3 - Q_1 = 31.75 - 11.75 = 20$

(g) Upper and Lower Limits of the Boxplot:

The limits are calculated as:

1. Lower Limit: $Q_1 - 1.5 \times IQR$ $\text{Lower Limit} = 11.75 - 1.5 \times 20 = 11.75 - 30 = -18.25$

2. Upper Limit: $Q_3 + 1.5 \times IQR$ $\text{Upper Limit} = 31.75 + 1.5 \times 20 = 31.75 + 30 = 61.75$

Final Answers:

• (c) Q1 = 11.75
• (d) Median (Q2) = 26
• (e) Q3 = 31.75
• (f) IQR = 20
• (g) Upper Limit = 61.75, Lower Limit = -18.25

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Would you like more details on any step or explanation?
Here are 5 related questions:

1. How do you calculate specific percentiles of a dataset?
2. What is the difference between quartiles and percentiles?
3. How does IQR help detect outliers in a dataset?
4. Can the boxplot limits vary based on different datasets?
5. Why is interpolation used when locating quartile positions?

Tip: Always double-check dataset sorting before performing quartile calculations.