Statistics Shorts
Below are some statistical statements. Can you decide whether each is always, sometimes, or never true?
If they are sometimes true, give examples or conditions under which they are true and under which they are false. Are some statements usually true, or almost always true, or almost never true?
If they are always true or never true, give convincing reasons why that is the case.
Be sure to be clear about your statistical assumptions in each case.
It is just as likely to rain tomorrow as it is to rain the day after tomorrow.
In a school, there will be two people who share a birthday.
In a school, it will be someone's birthday every day.
A randomly selected person from London will live to a greater age than a randomly selected person from Calcutta.
If everyone in the class rolled two dice until someone threw a double six, there would be one winner.
If I roll a die 100 times, I will get about the same number of 1s as 6s.
If I flip a fair coin 20 times, I will get 10 heads.
The mean, median and mode of a set of numbers can't all be the same.
The mean cannot be less than both the median and the mode.
Half of the students taking a test score less than the average mark.
Nobody scores higher than the average mark in a test.
In a game where you can only score an even number of points (0, 2, 10 or 50), the average score over a series of games must be an even number.
Could you adapt any of the statements that are sometimes true to make them always or never true?
Sometimes true. This statement is usually true since the possibility of it raining tomorrow, which is approximate, should be the same as the possibility of it raining the day after tomorrow. Nevertheless, there are always exceptions in weather predictions, meaning that this statement will not be always true.
Theoretically always true. This statement is always true because within 30 people the birthday matches is 435 pairs (29+28+27+...+2+1=30*29/2=435) which is a lot more than 366. Nevertheless, in reality the fact is that people’s birthdays could scattered, so in reality it is sometimes true, theoretically it is always true.
Sometimes true. This statement is almost never true. Suppose the amount of students in the school is more that 366, it is still hard for everyday to be a birthday because there might be several people’s birth on one single day. If the school has more than 366 students, and if there is a birthday everyday, the chance of the statement is 1/365!, which is a lot larger than the chance of the statement. However the more students the school has, the more likely the statement is possible.
Theoretically always true, since the life expectancy for UK is 80.1 while the life expectancy for india 66.8. So picking someone from London will probably live longer than someone picked in Calcutta. However it is possible to pick someone who might not get so good a health care and so in reality, the statement is usually true.
Theoretically never true. This is because the chance for everyone to get a double six is the same, so theoretically, when one student get a double six, every should get a double six as well. However, the possibility could never predict the future, meaning that in reality, it is usually always because it is going to be very hard to meet the statement’s standards especially when there are many students.
Theoretically always true. This is statement is theoretically always true because the chance of getting a 1 would 1/6 and the chance for getting a 6 would also be 1/6, so the chance is the same, but in real life probabilities cannot predict for sure what is going to happen, so this statement is, in real life, sometimes true.
Theoretically always true. This statement is theoretically always true because the chance of getting a head when flipping a fair coin is 50% and so is the chance of getting a tail. Nevertheless, this is the theoretical probabilities and since we cannot predict the future and be very exact and accurate, so in reality, it is sometimes true.
Sometimes true. Suppose that there is a set of numbers: 3, 3, and 3. The mean is 3; the median is 3; and the mode is 3 as well, so it is possible for The mean, median and mode of a set of numbers to be all the same. But it is also possible for them to be different. In a set of numbers: 1, 3, 5, 10, and 10, the mean is 5.8; the median is 5; and the mode is 10. In short, this statement is sometimes true, and it is usually always true.
Sometimes true. In a set of numbers: 3, 3, and 3, the mean is 3; the median is 3; and the mode is 3 as well, so the mean is not less than, but equal to the median and the mode. But this statement is also possible. In a set of numbers: 1, 2, 5, 5, and 5, the mean is; the mode is 5; and the median
Sometimes true. If these are 10 students in a class and if the scores they got are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the average score would be 10x(1+10)/2/10, which is 5.5. And 5 students are under the average; and if there are still 10 students and the scores they got are 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, the average would be (1+2+2+2+2+2+2+2+2+3)/10, which is (2x10)/10, which is 2; and only 1 student’s score less than the average. In conclusion this is possible but the chance is almost
Sometimes true. This statement is almost never true because the only scenario in which the statement is possible is when all the students got the exactly same score. For example, in a class of 10 students, if everyone got 5 points, the average would be 5 and nobody has got a score higher than that. But usually, it is rarely possible.
Always true. An n amount of 2k’s added together and divided by n is definitely going to be an even number. (2k+2k+…..2k)/n=[(2k)n]/n=2k, which is an even number.