Part a-1: Construct the relative frequency distribution

The relative frequency of each interval can be calculated by dividing the frequency of each interval by the total number of observations.

Step 1: Sum of frequencies

Total frequency $T = 36 + 39 + 49 + 28 = 152$

Step 2: Relative Frequency Calculation

• For $10 < x \leq 20$:
  $\text{Relative frequency} = \frac{36}{152} = 0.237$
• For $20 < x \leq 30$:
  $\text{Relative frequency} = \frac{39}{152} = 0.257$
• For $30 < x \leq 40$:
  $\text{Relative frequency} = \frac{49}{152} = 0.322$
• For $40 < x \leq 50$:
  $\text{Relative frequency} = \frac{28}{152} = 0.184$

Relative Frequency Distribution:

Interval	Frequency	Relative Frequency
$10 < x \leq 20$	36	0.237
$20 < x \leq 30$	39	0.257
$30 < x \leq 40$	49	0.322
$40 < x \leq 50$	28	0.184

Part a-2: Proportion of observations in the interval $20 < x \leq 30$

The proportion of observations in the interval $20 < x \leq 30$ is the relative frequency of that interval:

$\text{Proportion} = 0.257$

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Part b-1: Cumulative frequency distribution and cumulative relative frequency distribution

To construct the cumulative frequency and cumulative relative frequency distribution, we add up frequencies and relative frequencies as we move down the intervals.

Cumulative Frequency:

• $10 < x \leq 20$: $36$
• $20 < x \leq 30$: $36 + 39 = 75$
• $30 < x \leq 40$: $75 + 49 = 124$
• $40 < x \leq 50$: $124 + 28 = 152$

Cumulative Relative Frequency:

• $10 < x \leq 20$: $\frac{36}{152} = 0.237$
• $20 < x \leq 30$: $\frac{75}{152} = 0.493$
• $30 < x \leq 40$: $\frac{124}{152} = 0.816$
• $40 < x \leq 50$: $\frac{152}{152} = 1.000$

Cumulative Frequency and Relative Frequency Distribution:

Interval	Cumulative Frequency	Cumulative Relative Frequency
$10 < x \leq 20$	36	0.237
$20 < x \leq 30$	75	0.493
$30 < x \leq 40$	124	0.816
$40 < x \leq 50$	152	1.000

Part b-2: Number of observations that are 40 or less

The number of observations that are 40 or less is the cumulative frequency for the interval $30 < x \leq 40$:

$\text{Observations} = 124$

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Part c: Proportion of observations that are 40 or less

The proportion of observations that are 40 or less is the cumulative relative frequency for the interval $30 < x \leq 40$:

$\text{Proportion} = 0.816 \approx 0.8$

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Would you like more detailed explanations or any clarifications on these steps?

Here are some additional related questions:

1. How would the relative frequency change if the frequency in the second interval doubled?
2. What is the difference between cumulative frequency and relative frequency?
3. Can cumulative relative frequency ever exceed 1.000? Why or why not?
4. How can we visually represent the cumulative frequency distribution?
5. What does it mean if two intervals have the same relative frequency?

Tip: Cumulative frequency and cumulative relative frequency are useful for identifying patterns over a range of values, giving insight into the distribution of data across intervals.