Statistics revision - 2

The average (arithmetic mean) number of passengers on a subway car is 60. If the number of passengers on a car has a normal distribution with a standard deviation of 20, approximately what percent of subway cars carry lesser than 80 passengers?

On a particular test whose scores are distributed normally, the 2nd percentile is 1720, while the 84th percentile is 1990. What score, rounded to the nearest 10, most closely corresponds to the 16th percentile?

• 1750

• 1770

• 1790

• 1810

• 1830

Answer explanation

mean - 2(s.d) = 1720

mean + s.d = 1990

Solving this system of linear equations, we get, mean = 1900.

Hence s.d = 90.

Hence, mean - s.d = 1810.

In a class, there are two students whose score details are given below.

The first student's score is in the 68th percentile. The second student's score is in the 32nd percentile.

Quantity A: The percentage of the class that has scored lesser than the first student.

Quantity B: The percentage of the class that has scored more than the second student.

• A

• B

• C

• D

On a given math test out of 100 points, the vast majority of the 149 students in a class scored either a perfect score or a zero, with only one student scoring within 5 points of the mean. Which of the following logically follows about Set T, made up of the scores on the test?

• Set T will not be normally distributed

• The range of Set T would be significantly smaller if the scores had been more evenly distributed.

• The mean of Set T will not equal the median.

Answer explanation

Range is just largest - smallest value. It has got nothing to do with the way the numbers are distributed.

It may so happen that the student's score which is within 5 points of the mean is equal to the median value.

Home values among the 8,000 homeowners of Town X are normally distributed, with a standard deviation of $11,000 and a mean of $90,000.

Quantity A: The number of homeowners in Town X whose home value is above $112,000.

Quantity B: 300

• A

• B

• C

• D

Answer explanation

Given: Mean = 90,000.

Standard deviation = 11,000.

Note that 112,000 = 90,000 + 2(11,000).

We want values that are higher than mean + 2(S.d).

Since the data is normally distributed, we want 2% of the entire data (8,000 homes).

Hence, A = 200.

Set X consists of 9 total terms, but only two different terms. Six of the terms are each equal to twice the value of each of the remaining 3. Which of the following would provide sufficient additional information to determine the average of the set?

Indicate all such statements.

• The smaller number is positive and is 3 less than the larger number.

• The biggest term in the set is 6.

Answer explanation

Let the larger term be y and the smaller term be x.

Given: y = 2x.

Since there are 6 of the larger terms and 3 of the smaller ones, the average is { 6(2x) + 3(x) } / 9.

This is 15x/9. If we know, x or y, we can figure out the average.

Knowing standard deviation will not help find the individual values of x or y as standard deviation is just a measure of the average distance between point and the mean of the data.

Set S is a set of distinct positive integers. The standard deviation of Set S must increase if which of the following were to occur?

Indicate all such statements.

• Each number in the set is multiplied by 1/2.

• The smallest number is increased to become equal to the median

• The smallest number is increased to become larger than the current largest number.

• The largest number is doubled.

Answer explanation

Statement 1: In a set of distinct (different) integers, if each number is multiplied by 1/2, the numbers will get closer together (for instance, 2, 4, 6 would become 1, 2, 3), so the standard deviation would decrease.

Statement 2: If the smallest number in a set became equal to the median, then two numbers in the set would now be the same. The set would become less spread out, not more.

Example: Old set = { 1, 2, 3}. New set = {2, 2, 3}.

Statement 3: If the smallest number were increased to become larger than the current largest number, the standard deviation could increase, but not necessarily! For instance, if the set were 1, 2, 3, and the 1 were increased to become 100, the standard deviation would increase. But if the set were 1, 2, 3, and the 1 were increased to become 4, there will be no change in the spread at all.

What is the median of the data?

• 10

• 12

• 13

• 14

• 18

What is the range of the data?

• 80

• 10

• 70

• 40

• 30

What percent of the data falls between 8 and 10 absences?

• 0%

• 25%

• 50%

• 75%

• 100%