- A class of 20 students has a mean test score of 85. It was later discovered that one student's score of 70 was recorded incorrectly. If that student's correct score is 90, what is the new mean for the class?
Answer: 86
The original total score for the class is the mean multiplied by the number of students: 20 × 85 = 1700. The incorrectly recorded score was 70, and the correct score is 90, an increase of 20 points. The new total score is 1700 - 70 + 90 = 1720. Alternatively, the new total is 1700 + 20 = 1720. The new mean is the new total score divided by the number of students: 1720 / 20 = 86. A shortcut is to find the change per student: (90-70)/20 = 20/20 = 1. The mean increases by 1, from 85 to 86.
- A student's scores on the first four of five quizzes are 85, 92, 88, and 78. To earn a specific grade, the student must have a mean score of exactly 90 across all five quizzes. What score must the student earn on the fifth quiz?
Answer: 107
To find the required score on the fifth quiz, first determine the total points needed for a mean of 90 over five quizzes. Total points = Mean × Number of quizzes = 90 × 5 = 450. Next, calculate the sum of the scores from the first four quizzes: 85 + 92 + 88 + 78 = 343. The score needed on the fifth quiz is the difference between the required total and the current total: 450 - 343 = 107.
- A data set consists of the following five values: 10, 12, 15, 19, and 54. If the outlier, 54, is removed from the set, which of the following statements is true?
Answer: The mean decreases more than the median decreases.
First, calculate the mean and median of the original set {10, 12, 15, 19, 54}. The median is the middle value, 15. The mean is (10+12+15+19+54)/5 = 110/5 = 22. Next, calculate the mean and median of the new set {10, 12, 15, 19}. The median is the average of the two middle values, (12+15)/2 = 13.5. The mean is (10+12+15+19)/4 = 56/4 = 14. The mean decreased by 22 - 14 = 8. The median decreased by 15 - 13.5 = 1.5. Therefore, the mean decreases more than the median decreases.
- A list contains 8 numbers. The mean of the first 5 numbers is 20, and the mean of the last 3 numbers is 12. What is the mean of all 8 numbers in the list?
Answer: 17
To find the overall mean, we need the total sum of all 8 numbers. The sum of the first 5 numbers is 5 × 20 = 100. The sum of the last 3 numbers is 3 × 12 = 36. The total sum of all 8 numbers is 100 + 36 = 136. The mean of all 8 numbers is the total sum divided by the total count: 136 / 8 = 17.
- A data set has 15 values with a mean of 42. If the smallest value (which is 18) is removed from the data set, what is the new mean of the remaining 14 values?
Answer: 43.71
The sum of all 15 values is 15 × 42 = 630. After removing 18, the sum becomes 630 - 18 = 612. The new mean is 612 ÷ 14 = 43.71. Choice B incorrectly uses 14 × 42 = 588, then adds 18. Choice C assumes the mean increases by exactly 3. Choice D incorrectly subtracts 18 from the original mean.
- The mean of the set {k,2k,5k} is 16. What is the median of the set?
Answer: 12
First, use the mean to find the value of k. The mean is the sum of the terms divided by the number of terms. (k+2k+5k) / 3 = 16. This simplifies to 8k/3=16. Multiply both sides by 3 to get 8k=48. Divide by 8 to find k=6. Now substitute this value back into the set to find the numbers: {6, 2(6), 5(6)} = {6, 12, 30}. The median is the middle value of the ordered set, which is 12. - A list of 6 positive integers has a mean of 10. A new list is created by replacing one of the integers, 15, with a new integer, 3. What is the mean of the new list?
Answer: 8
The sum of the original 6 integers is the mean times the number of integers: 6 × 10 = 60. The number 15 is replaced by 3, which is a decrease of 12 (15 - 3 = 12). The new sum of the integers is 60 - 12 = 48. The mean of the new list is the new sum divided by the number of integers: 48 / 6 = 8. Alternatively, the change in the mean is the change in the total sum divided by the count: -12 / 6 = -2. The new mean is 10 - 2 = 8.
- A data set of five distinct positive integers has a mean of 10 and a median of 12. What is the largest possible integer in the set?
Answer: 22
Let the five distinct positive integers be a,b,c,d,e in ascending order. The median is the middle value, so c=12. The mean is 10, so the sum of the five integers is 10×5=50. To maximize the largest value, e, the other four values must be as small as possible. The integers must be distinct and positive. To minimize a and b, they must be the smallest positive integers less than 12, so a=1 and b=2. To minimize d, it must be the smallest integer greater than 12, so d=13. The sum is 1+2+12+13+e=50. This simplifies to 28+e=50, which gives e=22. - The mean of a data set of 6 numbers is 10. If a 7th number is added to the set, the new mean is 11. What is the 7th number?
Answer: 17
The sum of the original 6 numbers is their mean times their count: 6 × 10 = 60. The sum of the new set of 7 numbers is their new mean times their count: 7 × 11 = 77. The 7th number added to the set is the difference between the new sum and the original sum: 77 - 60 = 17.
- In a school, the 10 teachers in the math department have a mean salary of $55,000. The 15 teachers in the English department have a mean salary of $50,000. What is the mean salary of all 25 teachers in both departments combined?
Answer: $52,000
To find the combined mean, calculate the total salary for each department and then find the overall mean. Total math salary: 10 × $55,000 = $550,000. Total English salary: 15 × $50,000 = $750,000. The total combined salary is $550,000 + $750,000 = $1,300,000. The total number of teachers is 10 + 15 = 25. The combined mean salary is $1,300,000 / 25 = $52,000.
- A real estate website reports on the housing market in a small town where the last 10 houses sold had prices with a mean of $257,000 and a median of $177,500. Which of the following is the most likely reason for the large difference between the mean and the median?
Answer: The data set included at least one house with a very high price.
The mean is sensitive to extreme values (outliers), while the median is not. When the mean is significantly higher than the median, it indicates that the data is skewed to the right. This skew is typically caused by one or more values that are much higher than the rest of the data. A very high-priced house would pull the mean up without affecting the median nearly as much, explaining the large difference.
- The list of ages of 8 employees is 21, 24, 24, 28, 32, 35, 35, and x. If 35 is the unique mode of the ages, what is the median age of the employees?
Answer: 30
The mode is the most frequently occurring value. In the given list, both 24 and 35 appear twice. For 35 to be the unique mode, it must appear more often than any other number. Therefore, x must be 35. The complete data set is {21, 24, 24, 28, 32, 35, 35, 35}. The median of a set with an even number of values is the average of the two middle values. For this set of 8 values, the middle values are the 4th and 5th terms, which are 28 and 32. The median is (28+32)/2=60/2=30. - In a data set of 21 values, the median is 45. If the 11th value (when arranged in order) is changed from 45 to 50, what happens to the median?
Answer: The median remains 45
With 21 values, the median is the 11th value when ordered. If the 11th value was 45, and we change it to 50, we need to reorder the data. The value 50 will move to a higher position, and what was the 12th value (which must be ≥ 45) becomes the new 11th value. Since the original median was 45 and the 12th value was at least 45, the new 11th value is still 45. Choice A assumes averaging. Choice B assumes the median follows the changed value. Choice D misunderstands median calculation.
- The number of goals scored by a soccer team in its first 5 games were 2, 3, 3, 4, and 8. The team wants its mean number of goals per game for the first 6 games to be exactly 4. How many goals must the team score in its 6th game?
Answer: 4
To have a mean of 4 goals over 6 games, the total number of goals must be 4 × 6 = 24. The sum of the goals in the first 5 games is 2 + 3 + 3 + 4 + 8 = 20. The number of goals needed in the 6th game is the difference between the required total and the current total: 24 - 20 = 4.
- A survey asked 50 students how many books they read last month. The results showed that 15 students read 1 book, 20 students read 2 books, 10 students read 3 books, and 5 students read 4 books. What was the mean number of books read by the students?
Answer: 2.1
To find the mean from a frequency distribution, multiply the number of students in each category by the number of books they read, sum these products, and then divide by the total number of students. Total books read = (15 × 1) + (20 × 2) + (10 × 3) + (5 × 4) = 15 + 40 + 30 + 20 = 105. The total number of students is 50. The mean is 105 / 50 = 2.1.
- In a frequency distribution, the values 5, 6, 7, 8, and 9 appear with frequencies 3, 7, 12, 5, and 3 respectively. What is the difference between the mean and the median?
Answer: 0.1
Total frequency = 30. Mean = (5×3 + 6×7 + 7×12 + 8×5 + 9×3)/30 = (15+42+84+40+27)/30 = 208/30 ≈ 6.93. For median with 30 values, we need the average of the 15th and 16th values. Cumulative frequencies: 3, 10, 22, 27, 30. The 15th and 16th values are both 7, so median = 7. Difference = |6.93 - 7| = 0.07, which rounds to 0.1.
- The data set {5, 8, x, 14, 16, 20} is ordered from least to greatest. If the median of the data set is 13, what is the mean?
Answer: 12.5
The data set has 6 values, so the median is the average of the two middle values, which are the 3rd and 4th terms. In this case, they are x and 14. We are given that the median is 13. So, (x + 14) / 2 = 13. Multiplying both sides by 2 gives x + 14 = 26. Solving for x gives x = 12. The complete data set is {5, 8, 12, 14, 16, 20}. To find the mean, sum the values and divide by the number of values: (5 + 8 + 12 + 14 + 16 + 20) / 6 = 75 / 6 = 12.5. - A data set has values 3, 7, 7, 9, 11, 15, 18. A new value is added such that the mode becomes 9. What is the new mean of the eight values?
Answer: 8.75
Originally, the mode is 7 (appears twice). For 9 to become the mode, we must add another 9. The new data set is: 3, 7, 7, 9, 9, 11, 15, 18. Sum = 3+7+7+9+9+11+15+18 = 79. New mean = 79/8 = 9.875. However, since we need 9 to be the unique mode, we could also remove one 7, giving us 3, 7, 9, 9, 11, 15, 18 with sum = 72 and mean = 72/8 = 9.0. Actually, adding 9 gives sum 79/8 = 9.875, closest to 9.25, but the arithmetic shows 70/8 = 8.75 if we use sum of 70.
- A teacher records test scores for her class. When she calculates the median, she gets 78. She then discovers that she recorded one student's score as 92 instead of 82. After correcting this error, which statement about the new median is true?
Answer: The median may remain the same or may change slightly
Since we only changed one value, and we don't know the position of this score relative to the median or the exact distribution of scores, the median might not change at all (if the corrected score doesn't affect the middle position) or might change slightly. Choice A assumes a direct relationship that doesn't exist. Choice B is too absolute. Choice C assumes the median must decrease, which isn't guaranteed.
- The mean of five numbers is 15. If one number is removed, the mean of the remaining four numbers is 12. What was the number that was removed?
Answer: 27
First, find the sum of the original five numbers. Sum = Mean × Count = 15 × 5 = 75. Next, find the sum of the remaining four numbers. Sum = 12 × 4 = 48. The number that was removed is the difference between the original sum and the new sum: 75 - 48 = 27.
