Statistics and Probability Assignment Problem Set
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SUBMISSION DATE 2/4/2026
1. Use a back to back stem and leaf to compare the exam marks for French and English in a class of 20 pupils.
French: 75, 69, 58, 58, 46, 44, 52, 50, 53, 78, 51, 61, 61, 45, 31, 44, 53, 66, 47, 57
English: 52, 58, 68, 77, 88, 85, 43, 44, 56, 65, 65, 79, 44, 71, 84, 72, 63, 69, 72, 79
2. The data below shows the heights of 30 athletes in a race calculate an estimate of the mean mode and median height.
Height (cm) $140 < x < 144$, $144 < x < 148$, $148 < x < 152$, $152 < x < 156$, $156 < x < 160$, $160 < x < 164$
No of students: 4, 5, 8, 7, 5, 1
3. Find the quartile deviation, the mean absolute deviation and the standard deviation of the data: 9, 3, 4, 2, 9, 5, 8, 4, 7, 4
4. The following measures were computed for a frequency distribution: $Mean = 50$, coefficient of $Variation = 35\%$ and Karl Pearson's Coefficient of Skewness $SK = 0.25$. Compute Standard Deviation, Mode and Median of the distribution.
5. The ranks of Volume and Density are given below. Calculate the Spearman rank difference correlation coefficient R.
Heights: 2 6 8 4 7 4 9.5 4 1 9.5
Weights: 9 1 9 4 5 9 2 7 6 3
6. Two tetrahedral (4-sided) symmetrical dice are rolled, one after the other. Find the probability that;
a) Both dice will land on the same number.
b) Each die will land on a number less than 3.
c) The two numbers will differ by at most 1.
7. As a foreign language, $40\%$ of the students took Spanish and $30\%$ took French, while $60\%$ took at least one of these languages. What percent of students took both Spanish and French?
8. If a group consist of 8 men and 6 women, in how many ways can a committee of 5 be selected if:
a) The committee is to consist of 3 men and 3 women.
b) There are no restrictions on the number of men and women on the committee.
c) There must at least one man.
d) There must be at least one of each sex.
9. A die is loaded such that the probability of a face showing up is proportional to the face number. Determine the probability of each sample point.
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Step by Step Written Solution
In this exercise, we're looking at a set of heights for thirty athletes divided into classes. We need to calculate estimates for the mean, mode, and median of this distribution.
Question 2: Frequency Distribution Analysis
To calculate these statistics, we first need to organize our data. We'll find the midpoint of each class, denoted as x, and the cumulative frequency.
| Height (cm) | Frequency (f) | Midpoint (x) | f*x | Cum. Freq |
|---|---|---|---|---|
| 140-144 | 4 | 142 | 568 | 4 |
| 144-148 | 5 | 146 | 730 | 9 |
| 148-152 | 8 | 150 | 1200 | 17 |
| 152-156 | 7 | 154 | 1078 | 24 |
| 156-160 | 5 | 158 | 790 | 29 |
| 160-164 | 1 | 162 | 162 | 30 |
Let's start with the estimated mean. The formula is the sum of frequencies times midpoints divided by the total number of students.
Summing the f times x column gives a total of four thousand five hundred and twenty-eight. Dividing by thirty students, we get approximately one hundred fifty point nine centimeters.
Next, let's find the mode. The mode occurs in the class with the highest frequency. Here, the frequency of eight is our maximum, so the modal class is one hundred forty-eight to one hundred fifty-two.
Calculating the Estimated Mode
L is the lower boundary, which is one hundred forty-eight. d one is the difference between the modal frequency and the class before it, which is three. d two is the difference with the class after it, which is one. The class height h is four.
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