Answer :
Problem 8
Let's calculate the mean, median, mode, and standard deviation of the test scores provided.
Given Scores and Frequencies:
- 90: Frequency = 1
- 80: Frequency = 3
- 70: Frequency = 10
- 60: Frequency = 5
- 50: Frequency = 2
1. Mean
The mean is calculated by multiplying each score by its frequency, summing these products, and dividing by the total number of observations.
[tex]\text{Mean} = \frac{(90 \times 1) + (80 \times 3) + (70 \times 10) + (60 \times 5) + (50 \times 2)}{1 + 3 + 10 + 5 + 2}[/tex]
[tex]= \frac{90 + 240 + 700 + 300 + 100}{21}[/tex]
[tex]= \frac{1430}{21} \approx 68.10[/tex]
2. Median
The median is the middle value when scores are ordered. With 21 scores, the median is the 11th value.
Scores in order:
50, 50, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70, 70, 70, 80, 80, 80, 90
The 11th value is 70, so the median is 70.
3. Mode
The mode is the most frequently occurring value, which is 70 (10 times).
4. Standard Deviation (Sample)
First, calculate each score's deviation from the mean, square it, multiply by frequency, sum them, and divide by degrees of freedom (n-1 = 20).
[tex]\text{Variance} = \frac{\sum (x_i - \text{mean})^2 \cdot f}{n-1}[/tex]
After calculating,
Variance ≈ 88.95
Standard Deviation = [tex]\sqrt{88.95} \approx 9.43[/tex]
Problem 9
Percentile Rank:
The percentile rank of a value could be found using the formula:
[tex]\text{Percentile Rank} = \frac{(B + 0.5E)}{N} \times 100[/tex]
where:
- [tex]B[/tex] is the number of scores below the given score,
- [tex]E[/tex] is the total count of scores equal to the given score,
- [tex]N[/tex] is the total number of scores.
(a) 14th out of 75:
(
\text{Percentile Rank} = \frac{14}{75} \times 100 = 18.67
)
(b) 83rd out of 690:
(
\text{Percentile Rank} = \frac{83}{690} \times 100 \approx 12.03
)
(c) 106th out of 17,500:
(
\text{Percentile Rank} = \frac{106}{17500} \times 100 \approx 0.606
)
Problem 10
Class Rank from Percentile:
The class rank can be found using the inverse relationship of percentile.
(a) For 55th percentile out of 140:
(
\text{Rank} = \frac{55}{100} \times 140 = 77
)
(b) For 99th percentile out of 5,000:
(
\text{Rank} = \frac{99}{100} \times 5000 = 4950
)
(c) For 70th percentile out of 381:
(
\text{Rank} = \frac{70}{100} \times 381 \approx 266.7 \approx 267
)
Problem 11
Given Examination Scores:
{93, 86, 83, 75, 71, 67, 65, 63, 60, 53, 45, 48}
Let's rearrange the scores in order: 45, 48, 53, 60, 63, 65, 67, 71, 75, 83, 86, 93
(a) Percentile rank of score 93:
93 is the highest score. The rank of 93 is:
(
\text{Percentile rank} = \frac{12}{12} \times 100 = 100
)
(b) Median:
- Middle two numbers are 65 and 67.
- (
\text{Median} = \frac{65 + 67}{2} = 66
)
(c) Third Quartile (Q3):
Q3, the 75th percentile, is the 9th score, which is 75.
(d) 70th percentile score:
Approximately the 8th score in the list, which is 71.
Problem 12
Probability of Choosing a Clear Marble:
Given:
- Blue marbles = 25
- Cat-eye marbles = 55
- Clear marbles = 20
- Total marbles = 25 + 55 + 20 = 100
Probability of clear marble:
(
\text{Probability} = \frac{\text{Number of clear marbles}}{\text{Total marbles}} = \frac{20}{100} = 0.2
)
Thus, the probability of drawing a clear marble is [tex]0.2[/tex] or [tex]20\%[/tex].
