Question 1184639: Please help, I've been trying to find out how to do these problems online for hours.
For the following distribution of scores:
x f
10 1
9 2
8 4
7 5
6 3
5 1
4 1
3 1
a) Find the middle 30% of the scores.
b) Find the bottom 20% of the scores.
c) Find the top 10% of the scores.
d) Find the bottom 70% of the scores.
e) Find the top 60% of the scores.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! You're right, these percentile problems can be tricky! Let's break down each one step by step, and I'll explain the logic clearly.
**1. Calculate Cumulative Frequencies:**
This is the crucial first step. Cumulative frequency tells you how many scores are *at or below* a given value.
| x | f | Cumulative f |
| --- | --- | ------------- |
| 10 | 1 | 1 |
| 9 | 2 | 3 |
| 8 | 4 | 7 |
| 7 | 5 | 12 |
| 6 | 3 | 15 |
| 5 | 1 | 16 |
| 4 | 1 | 17 |
| 3 | 1 | 18 |
The total number of scores (N) is 18.
**2. Understanding Percentiles:**
A percentile tells you the score at or below which a certain percentage of the data falls. For example, the 20th percentile is the score at or below which 20% of the scores lie.
**3. The Calculations:**
**a) Middle 30%:**
* **Logic:** The middle 30% is *between* the 35th and 65th percentiles.
* **35th Percentile:** (0.35) * 18 = 6.3. Since this isn't a whole number, we round *up* to the nearest whole number, which is 7. The 7th score in the cumulative frequency table is 8.
* **65th Percentile:** (0.65) * 18 = 11.7. We round *up* to 12. The 12th score is 7.
* **Answer:** The middle 30% of scores are 7 and 8.
**b) Bottom 20%:**
* **Logic:** We're looking for the score at or below which 20% of the data falls.
* **(0.20) * 18 = 3.6. Round up to 4.** The 4th score in the cumulative frequency table is 8.
* **Answer:** The bottom 20% of the scores are 8 and below. (It's a little counterintuitive because the *value* is 8, but it represents the *cutoff* for the bottom 20% of the *scores*.)
**c) Top 10%:**
* **Logic:** This is the score at or above which 10% of the data falls. This is the same as the 90th percentile (100% - 10% = 90%).
* **(0.90) * 18 = 16.2. Round up to 17.** The 17th score is 4.
* **Answer:** The top 10% of the scores are 4 and above.
**d) Bottom 70%:**
* **(0.70) * 18 = 12.6. Round up to 13.** The 13th score is 7.
* **Answer:** The bottom 70% of the scores are 7 and below.
**e) Top 60%:**
* **Logic:** This is the same as the 40th percentile (100% - 60% = 40%).
* **(0.40) * 18 = 7.2. Round up to 8.** The 8th score is 8.
* **Answer:** The top 60% of the scores are 8 and above.
**Key Points to Remember:**
* **Round Up:** When the percentile calculation results in a decimal, *always* round up to the next whole number. This is because we're looking for the *position* of the score in the ordered list.
* **Cumulative Frequency:** The cumulative frequency table is essential for these problems.
* **"At or Below/Above":** Percentiles represent the value *at or below* a certain percentage of the data. So, the bottom 20% includes all scores up to and including the score at the 20th percentile.
I hope this detailed explanation helps! Let me know if you have any more questions.
|
|
|
