High School students in a100- item test $$ 82,45,37,98,100,74,87,89,63 $$ $$ 76,75,61,43,99 $$ 5. The midterm score of 45 is the. $$ \qquad $$ decile. 7. The midterm score of 89 is the $$ \qquad $$ quartile. 3. The midterm score of 74 is the $$ \qquad $$ percentile. . The percentile ranking of 61 is $$ \qquad $$ percent. 10. If the passing score starts in 7.5th decile, $$ \qquad $$ tudents passed the test. 67.14 2nd 3rd 40th 21.438 1st

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To solve the problem, we need to analyze the given scores and calculate the required statistical measures: deciles, quartiles, percentiles, and percentile rankings. Let's break it down step by step.

### Given Scores
The scores are:
$$ 
82, 45, 37, 98, 100, 74, 87, 89, 63, 76, 75, 61, 43, 99 
$$

### Step 1: Sort the Scores
First, we need to sort the scores in ascending order:
$$ 
37, 43, 45, 61, 63, 74, 75, 76, 82, 87, 89, 98, 99, 100 
$$

### Step 2: Calculate Deciles
Deciles divide the data into ten equal parts. The $$ k $$-th decile can be found using the formula:
$$
D_k = \frac{k(n + 1)}{10}
$$
where $$ n $$ is the number of observations.

1. **For the midterm score of 45 (5th decile)**:
   - $$ n = 14 $$
   - $$ D_5 = \frac{5(14 + 1)}{10} = \frac{5 \times 15}{10} = 7.5 $$
   - The 5th decile is between the 7th and 8th scores: $$ 75 $$ and $$ 76 $$.
   - Therefore, the 5th decile is $$ 75.5 $$.

2. **For the midterm score of 89 (3rd quartile)**:
   - The 3rd quartile (Q3) is the 75th percentile, which corresponds to the 7.5th decile.
   - $$ D_7 = \frac{7(14 + 1)}{10} = \frac{7 \times 15}{10} = 10.5 $$
   - The 10th score is $$ 87 $$ and the 11th score is $$ 89 $$.
   - Therefore, the 3rd quartile is $$ 88 $$.

### Step 3: Calculate Percentiles
Percentiles divide the data into 100 equal parts. The $$ k $$-th percentile can be found using the formula:
$$
P_k = \frac{k(n + 1)}{100}
$$

1. **For the midterm score of 74 (40th percentile)**:
   - $$ P_{40} = \frac{40(14 + 1)}{100} = \frac{40 \times 15}{100} = 6 $$
   - The 6th score is $$ 74 $$.
   - Therefore, the 40th percentile is $$ 74 $$.

### Step 4: Calculate Percentile Ranking
The percentile ranking of a score is the percentage of scores that fall below that score.

1. **For the score of 61**:
   - There are 4 scores below 61: $$ 37, 43, 45, 61 $$.
   - The percentile ranking is:
   $$
   \text{Percentile Ranking} = \frac{\text{Number of scores below } 61}{n} \times 100 = \frac{4}{14} \times 100 \approx 28.57\%
   $$

### Step 5: Determine Passing Students
If the passing score starts in the 7.5th decile, we need to find the score at that decile.

1. **7.5th Decile**:
   - The 7.5th decile is between the 7th and 8th scores: $$ 75 $$ and $$ 76 $$.
   - Therefore, the passing score is $$ 75.5 $$.

2. **Count Students Passing**:
   - Scores above $$ 75.5 $$: $$ 76, 82, 87, 89, 98, 99, 100 $$ (7 students).
   - Therefore, $$ 7 $$ students passed the test.

### Final Answers
- The midterm score of 45 is the **5th decile**.
- The midterm score of 89 is the **3rd quartile**.
- The midterm score of 74 is the **40th percentile**.
- The percentile ranking of 61 is approximately **28.57%**.
- If the passing score starts in the 7.5th decile, **7** students passed the test.
- The midterm score of 45 is the 5th decile. - The midterm score of 89 is the 3rd quartile. - The midterm score of 74 is the 40th percentile. - The percentile ranking of 61 is approximately 28.57%. - 7 students passed the test.

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