Week Two Practice Problems Worksheet
Question 1: The average score of students in a class is 85 with a standard deviation of 10. If a new student joins the class and scores 95 on the exam, what is the new average score of the class?
To find the new average score of the class, we need to consider the impact of the new student’s score on the overall average.
The average score is calculated by summing up all the individual scores and dividing it by the total number of students. In this case, we are given that the average score of the class without the new student is 85. This means that the sum of all the individual scores (excluding the new student) is 85 multiplied by the total number of students.
Let’s assume the total number of students in the class without the new student is ‘n’. Therefore, the sum of all the individual scores before the new student joined is 85n.
Now, to calculate the new average score after the new student joined, we need to add the new student’s score to the sum of all the individual scores. The new sum of all the individual scores will be 85n + 95.
Since the new student has joined the class, the total number of students in the class is now ‘n + 1’. To find the new average score, we divide the new sum of all the individual scores by the total number of students.
Thus, the new average score of the class (after the new student joined) can be calculated as follows:
New average score = (85n + 95) / (n + 1)
Question 2: A company employs 100 people. The average salary is $40,000 with a standard deviation of $5,000. If the company decides to raise everyone’s salary by $2,000, what is the new average salary?
To find the new average salary of the company, we need to take into account the increase in salaries for all employees.
Since the average salary is calculated by summing up all the individual salaries and dividing it by the total number of employees, we can calculate the sum of all the individual salaries before the increase.
The sum of all the individual salaries before the increase is $40,000 multiplied by the total number of employees, which is $40,000 * 100 = $4,000,000.
Now, to calculate the new average salary after the increase, we need to add $2,000 to each individual salary. This means that the new sum of all the individual salaries will be $4,000,000 + ($2,000 * 100) = $4,200,000.
Since the company employs 100 people, the total number of employees remains the same. Therefore, to find the new average salary, we divide the new sum of all the individual salaries by the total number of employees.
Thus, the new average salary can be calculated as follows:
New average salary = $4,200,000 / 100 = $42,000.
Question 3: The average height of a population is 68 inches with a standard deviation of 3 inches. If a person’s height is 71 inches, what is their z-score?
A z-score is a measure of how many standard deviations an individual’s score is above or below the average score. It is calculated by subtracting the average score from the individual’s score and then dividing the result by the standard deviation.
In this case, the average height of the population is 68 inches, and the standard deviation is 3 inches. The person’s height is 71 inches.
Therefore, the z-score can be calculated as follows:
Z-score = (71 – 68) / 3 = 3 / 3 = 1.
The person’s height of 71 inches is 1 standard deviation above the average height of the population.
In conclusion, these practice problems involve calculating new averages, considering the impact of changes in individual scores or salaries on the overall average, and determining z-scores as measures of how individual scores deviate from the mean in a population. These concepts are fundamental in statistics and can be applied in various real-life scenarios.
