Applications to Data
We examine data in order to make inferences about the likelihood of particular outcomes in subsequent observations.
For example, a political pollster gathers survey data in order to predict the outcome of an election, or a manufacturer checks a sample of items from an assembly line in order to be confident that only a few of the products will fail to meet quality standards.
The information gleaned from the data tends to take the form of probabilities.
Example
Suppose $55$55 students complete a test. The students are ranked from $1$1 to $55$55 according to their scores in the test. The student in ranking position $28$28 is in the median position because there are $27$27 students with a lower ranking and $27$27 with a higher ranking.
Notice that half the students are at or above the median ranking and half are at or below, and this would be true no matter how carefully they had studied for the test. Each student has a probability of $0.5$0.5 of being ranked at least as high as the median student.
A more useful way of looking at the test results belonging to the $55$55 students would be to sort the scores into groups according to the actual scores rather than their ranking positions.
Suppose the median student achieved an actual score of $64%$64% on the test. There are many ways in which the remaining scores could be spread across the full range of scores. The following frequency table shows one possibility. We have split the scores into bins $10$10 units wide.
| score % | < 10 | 11-20 | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 | 81-90 | 91-100 |
|---|---|---|---|---|---|---|---|---|---|---|
| frequency | $1$1 | $1$1 | $2$2 | $7$7 | $6$6 | $10$10 | $14$14 | $9$9 | $3$3 | $2$2 |
The information from this frequency table can be displayed in the form of a column graph.
This diagram makes it clear that most of the 55 test scores were clustered in the range from 31% to 80%.
By counting the number of scores in these middle five bins we can work out the proportion of students who achieved scores in this range.
$\frac{7+6+10+14+9}{55}=\frac{37}{55}\approx0.67$7+6+10+14+955=3755≈0.67
We would then say that the probability of a student chosen at random from this group having a test score between $31%$31% and $80%$80% is $0.67$0.67.
In a similar way, we could work out that the probability of a randomly chosen student from this group having a test score of above $90%$90% is $\frac{2}{55}\approx0.036$255≈0.036.
Worked Examples
Question 1
The times from the 800m final at the 2016 Summer Olympics are given in the table:
| Rank | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th |
|
2016 Summer Olympics |
$1:42.15$1:42.15 | $1:42.61$1:42.61 | $1:42.93$1:42.93 | $1:43.41$1:43.41 | $1:43.55$1:43.55 | $1:44.20$1:44.20 | $1:46.02$1:46.02 | $1:46.15$1:46.15 | $1:45.24$1:45.24 |
If an athlete from the race was chosen at random, what is the probability that their time was faster than $1:43.55$1:43.55? Give your answer as a fraction.
Question 2
Junior's Garage conducts a stocktake of all replacement parts in their shop, recording the year of manufacture and quantity of each part.
The results are shown in the following table.
Complete the table, showing all totals.
Part 2008 2010 2012 2014 2016 Total number of parts EA5163b $22$22 $24$24 $29$29 $24$24 $21$21 $120$120 EF5149a $3$3 $6$6 $10$10 $5$5 $1$1 $25$25 EM3188f $15$15 $15$15 $17$17 $14$14 $0$0 $\editable{}$ Total number of parts $40$40 $45$45 $56$56 $\editable{}$ $22$22 $\editable{}$ If a part was found in the store, what is the probability that it is part number EM3188f?
Give your answer as a fraction in simplest form.
If a part was found in the store, what is the probability that it is newer than 2012?
Give your answer as a fraction in simplest form.
Question 3
Some people were asked approximately how many of their high school friends they remained in contact with after high school. The results were as follows:
$10,30,10,40,20,40,40,20,10,0,0,10,30,20,20,30,20,30,20,10,0,30,40,10,0,30,0,10,0,0,10,40,40,0,30,0,20,0,30,20,20,20$10,30,10,40,20,40,40,20,10,0,0,10,30,20,20,30,20,30,20,10,0,30,40,10,0,30,0,10,0,0,10,40,40,0,30,0,20,0,30,20,20,20
Complete the frequency table for this data set.
Score Frequency $0$0 $\editable{}$ $10$10 $\editable{}$ $20$20 $\editable{}$ $30$30 $\editable{}$ $40$40 $\editable{}$ How many people were surveyed in total?
If a person was chosen at random, what is the probability that they have kept in touch with $10$10 friends? Give your answer as a percentage to two decimal places where appropriate.
What is the probability that a person chosen at random kept in touch with less than $30$30 friends? Give your answer as a percentage to two decimal places where appropriate.