Statistics and probability: 99 IB Maths AI SL practice questions

The IB grades attained by a group of students are listed as follows.

6 4 5 3 7 3 5 4 2 5

(a) Find the median grade.[2]

(b) Calculate the interquartile range.[2]

(c) Find the probability that a student chosen at random from the group scored at least a grade 4.[2]

The following data show the heights, in metres, of six players in a basketball team.

(a) For these six players, find[6]

(a)(i) the mean height.

(a)(ii) the median height.

(a)(iii) the modal height.

(a)(iv) the range of the heights.

A new player, Gheorghe, joins the team. Their height is measured as 1.98 metres to the nearest centimetre.

(b) Write down the shortest possible height of Gheorghe.[1]

Consider the following list of 10 data items.

16 19 21 19 18 12 16 21 21 32

This data has a mean of 19.5.

Find the value of the

(a) mode.[1]

(b) median.[1]

(c) standard deviation.[1]

(d) range.[2]

A class of 13 Mathematics students received the following grades in their final IB examination: 3, 5, 3, 4, 7, 3, 2, 7, 5, 6, 5, 3, 4

For these grades, find:

(a) the mode;[1]

(b) the median;[2]

(c) the upper quartile;[1]

(d) the interquartile range.[2]

The prices, in dollars, of 10 different garden chairs are:

79, 139, 255, 99, 50, 209, 229, 193, 69, 49

(a) Find the range of the prices of the 10 chairs.[2]

(b) Use your graphic display calculator to find

(i) the mean price of the chairs.

(ii) the standard deviation of the price of the chairs.[3]

In a sale, the price of each of the 10 garden chairs is reduced by $20.

(c) Write down

(i) the new mean.

(ii) the new standard deviation.[2]

A tetrahedral (four-sided) die has written on it the numbers 1, 2, 3 and 4. The die is rolled many times and the scores are noted. The table below shows the resulting frequency distribution.

The die was rolled a total of 100 times.

(a) Write down an equation, in terms of $x$ and $y$, for the total number of times the die was rolled.[1]

The mean score is 2.71.

(b) Using the mean score, write down a second equation in terms of $x$ and $y$.[2]

(c) Find the value of $x$ and of $y$.[3]

A study was carried out to determine whether the country chosen by students for their university studies was influenced by a person's gender. A random sample was taken. The results are shown in the following table.

A $\chi^2$ test was performed at the 1% significance level. The critical value for this test is 9.210.

(a) State the null hypothesis.[1]

(b) Write down the number of degrees of freedom.[1]

(c) Write down[2]

(i) the $\chi^2$ statistic;

(ii) the associated $p$-value.

(d) State, giving a reason, whether the null hypothesis should be accepted.[2]

A fair game is played where points are scored as follows:

• A win scores $w$ points.

• A draw scores $0$ points.

• A loss scores $-5$ points.

Let $X$ be the number of points scored during a game. The probability distribution is shown.

(a) Find the value of $p$.[2]

The game is played 60 times.

(b) Find the expected number of losses.[2]

(c) Calculate the value of $w$, given that the game is fair.[3]

The following frequency distribution table shows the test grades for a group of students.

For this distribution, the mean grade is 4.5.

(a) Write down the total number of students in terms of $p$.[1]

(b) Calculate the value of $p$.[3]

The scores on a test, out of 7 points, for 220 students are shown in the following table.

(a) Find the mean and standard deviation of these test scores. Give your answers correct to four significant figures.[3]

(b) The scores are multiplied by ten. Write down the new mean and standard deviation.[2]

(c) After the scores have been multiplied by ten, 30 points are added to each of them. Write down the new mean and standard deviation.[2]

The fastest recorded speeds of eight animals are shown in the following table.

(a) State whether speed is a continuous or discrete variable.[1]

(b) Write down the median speed for these animals.[1]

(c) Write down the range of the animal speeds.[1]

(d) For these eight animals (i) find the mean speed; (ii) write down the standard deviation.[3]

A survey investigated the relationship between the number of cleaners, $n$, and the amount of time, $t$, it takes them to clean a school.

(a) Use your graphic display calculator to write down the equation of the regression line $t$ on $n$.[2]

(b) Write down the value of the Pearson's product–moment correlation coefficient, $r$.[2]

(c) Use your regression equation to find the amount of time 4 cleaners take to clean the school.[2]

A group of students were asked how long they spend practising mathematics during the week. The results are shown in the following table.

It is known that $35 < a < 52$.

(a) Write down[3]

(i) the modal class;

(ii) the mid-interval value of the modal class;

(iii) the class in which the median lies.

For this group of students, the estimated mean number of hours spent practising mathematics is 2.69.

(b) Calculate the value of $a$.[3]

The Home Shine factory produces light bulbs, 7% of which are found to be defective.

(a) Write down the probability that a light bulb produced by Home Shine is not defective.[1]

Francesco buys two light bulbs produced by Home Shine.

(b) (i) Find the probability that both light bulbs are not defective.[2]

(ii) Find the probability that at least one of Francesco's light bulbs is defective.[4]

The Bright Light factory also produces light bulbs. The probability that a light bulb produced by Bright Light is not defective is $a$.

Deborah buys three light bulbs produced by Bright Light.

(c) Write down an expression, in terms of $a$, for the probability that at least one of Deborah's three light bulbs is defective.[1]

The decathlon is a competition where athletes compete in ten events. Two of those events are long jump and high jump. In both events, a greater distance means a better ranking.

The table shows results for these two events at the World Championships.

The Spearman's rank correlation coefficient is used to determine if there is a linear correlation between an athlete's ranking in long jump and their ranking in high jump.

(a) Complete the table to show the athletes' rankings in high jump.[2]

(b) Find the value of the Spearman's rank correlation coefficient $r_s$.[2]

The following guide is used by the coach to determine the strength of the correlation between the ranks for long jump and high jump.

(c) State the strength of the correlation between the rankings as indicated by the table and interpret this in the context of the question.[2]