Question 1176654
Let's find the quartiles for the given grouped data.

**1. Understanding Quartiles**

* **Q1 (First Quartile):** The value below which 25% of the data falls.
* **Q2 (Second Quartile):** The median, the value below which 50% of the data falls.
* **Q3 (Third Quartile):** The value below which 75% of the data falls.

**2. Formulas for Grouped Data**

The formula for finding quartiles in grouped data is:

* Qk = L + [(kN/4 - cf) / f] * w

Where:

* Qk = the kth quartile
* L = lower boundary of the quartile class
* N = total number of data points
* cf = cumulative frequency of the class before the quartile class
* f = frequency of the quartile class
* w = class width

**3. Calculate Quartiles**

* N = 48
* Class width (w) = 5 (e.g., 50 - 46 + 1 = 5)

**a) Q1 (First Quartile)**

* kN/4 = (1 * 48) / 4 = 12
* The 12th value falls in the 26-30 class (cf = 19, which is > 12).
* L = 25.5
* cf = 8
* f = 11

* Q1 = 25.5 + [(12 - 8) / 11] * 5
* Q1 = 25.5 + (4 / 11) * 5
* Q1 = 25.5 + 20 / 11
* Q1 = 25.5 + 1.818
* Q1 ≈ 27.32

**b) Q2 (Second Quartile)**

* kN/4 = (2 * 48) / 4 = 24
* The 24th value falls in the 31-35 class (cf = 31, which is > 24).
* L = 30.5
* cf = 19
* f = 12

* Q2 = 30.5 + [(24 - 19) / 12] * 5
* Q2 = 30.5 + (5 / 12) * 5
* Q2 = 30.5 + 25 / 12
* Q2 = 30.5 + 2.083
* Q2 ≈ 32.58

**c) Q3 (Third Quartile)**

* kN/4 = (3 * 48) / 4 = 36
* The 36th value falls in the 36-40 class (cf = 42, which is > 36).
* L = 35.5
* cf = 31
* f = 11

* Q3 = 35.5 + [(36 - 31) / 11] * 5
* Q3 = 35.5 + (5 / 11) * 5
* Q3 = 35.5 + 25 / 11
* Q3 = 35.5 + 2.273
* Q3 ≈ 37.77

**Summary**

* Q1 ≈ 27.32
* Q2 ≈ 32.58
* Q3 ≈ 37.77