Statistics and probability: 72 IB Maths AI HL practice questions

The lengths of the seeds from a particular mango tree are approximated by a normal distribution with a mean of 4 cm and a standard deviation of 0.25 cm.

A seed from this mango tree is chosen at random.

(a) Calculate the probability that the length of the seed is less than 3.7 cm.[2]

It is known that 30% of the seeds have a length greater than $k$ cm.

(b) Find the value of $k$.[2]

For a seed of length $d$ cm, chosen at random, $\text{P}(4 - m < d < 4 + m) = 0.6$.

(c) Find the value of $m$.[2]

Taizo plays a game where he throws one ball at two bottles that are sitting on a table. The probability of knocking over bottles, in any given game, is shown in the following table.

(a) Taizo plays two games that are independent of each other. Find the probability that Taizo knocks over a total of two bottles.[4]

In any given game, Taizo will win $k$ points if he knocks over two bottles, win 4 points if he knocks over one bottle and lose 8 points if no bottles are knocked over.

(b) Find the value of $k$ such that the game is fair.[3]

Sergio is interested in whether an adult's favourite breakfast berry depends on their income level. He obtains the following data for 341 adults and decides to carry out a $\chi^2$ test for independence, at the 10% significance level.

(a) Write down the null hypothesis.[1]

(b) Find the value of the $\chi^2$ statistic.[2]

The critical value of this $\chi^2$ test is 7.78.

(c) Write down Sergio's conclusion to the test in context. Justify your answer.[2]

Jerry makes handcrafted chocolates. On average, 1 in 25 of the chocolates that Jerry makes is flawed. Whether or not a chocolate is flawed is independent of all other chocolates.

(a) In a batch of 20 chocolates, chosen at random, find the probability that[4]

(i) two are flawed.

(ii) more than two are flawed.

(b) Calculate the expected number of pesos Jerry makes from selling a batch of 20 randomly selected chocolates.[2]

Jerry sells the perfect chocolates for 50 pesos each and the flawed ones for 15 pesos each.

A group of 130 applicants applied for admission into either the Arts programme or the Sciences programme at a university. The outcomes of their applications are shown in the following table.

(a) Find the probability that a randomly chosen applicant from this group was accepted by the university.[1]

An applicant is chosen at random from this group. It is found that they were accepted into the programme of their choice.

(b) Find the probability that the applicant applied for the Arts programme.[2]

Two different applicants are chosen at random from the original group.

(c) Find the probability that both applicants applied to the Arts programme.[3]

Two judges, Brett and Clarence, rank the skill levels of eight sheepdogs in a competition. The sheepdogs are labelled A to H and the judges rank the dogs as shown in the table.

(a) Write down the rank that Brett awards sheepdog B.[1]

(b) Calculate Spearman's rank correlation coefficient for these data.[4]

(c) Comment on your answer to part (b) in terms of the ranks awarded by Brett and Clarence.[1]

Gustav plays a game in which he first tosses an unbiased coin and then rolls an unbiased six-sided die.

If the coin shows tails, the score on the die is Gustav's final number of points.

If the coin shows heads, one is added to the score on the die for Gustav's final number of points.

(a) Find the probability that Gustav's final number of points is 7.[2]

(b) Complete the following table.[3]

(c) Calculate the expected value of Gustav's final number of points.[2]

Jae Hee plays a game involving a biased six-sided die. The faces of the die are labelled $-3, -1, 0, 1, 2$ and $5$. The score for the game, $X$, is the number which lands face up after the die is rolled. The following table shows the probability distribution for $X$.

(a) Find the exact value of $p$.[1]

(b) Calculate the expected score.[2]

(c) Find the probability Jae Hee has a total score of $-3$.[3]

At the end of a school day, the Headmaster conducted a survey asking students in how many classes they had used the internet.

The data is shown in the following table.

(a) State whether the data is discrete or continuous.[1]

The mean number of classes in which a student used the internet is 2.

(b) Find the value of $k$.[4]

It was not possible to ask every person in the school, so the Headmaster arranged the student names in alphabetical order and then asked every 10th person on the list.

(c) Identify the sampling technique used in the survey.[1]

The temperatures at 5 different locations on a coral reef were measured. The mean of this sample was $20.1°C$ and the standard deviation of this sample, $s_n$, was $3.2°C$.

(a) Find an unbiased estimate of the population variance.[2]

(b) Assuming that the temperatures are normally distributed, find a 95% confidence interval for the mean temperature on the coral reef.[2]

(c) Using your answer to part (b), determine if it is plausible that the mean temperature on the coral reef could be $17°C$.[1]

A survey was answered by 20000 expatriates (people living in a country that is not their own). The data ranked countries in order of the country they felt was best for expatriates. The highest-ranked country was Switzerland.

These results were compared to happiness scores taken from *The World Happiness Report 2022*. The following table shows this data for the top 10 expatriate countries.

(a) For the happiness score, find[4]

(i) the upper quartile

(ii) the interquartile range.

(b) Show that Switzerland's happiness score is not an outlier for this data.[3]

The happiness scores were ranked to calculate Spearman's rank correlation coefficient, $r_s$. These ranks are shown in the following table.

(c) Write down the value of[3]

(i) $a$

(ii) $b$

(iii) $c$.

(d) (i) Find $r_s$.[3]

(ii) If France's happiness score is upgraded to 6.9, explain why the value of $r_s$ does not change.[3]

Jose concludes from this data that countries with high happiness scores are likely to be favourite expatriate countries.

(e) State, with a reason, whether Jose's conclusion is appropriate.[1]

A polygraph test is used to determine whether people are telling the truth or not, but it is not completely accurate. When a person tells the truth, they have a 20% chance of failing the test. Each test outcome is independent of any previous test outcome.

10 people take a polygraph test and all 10 tell the truth.

(a) Calculate the expected number of people who will pass this polygraph test.[2]

(b) Calculate the probability that exactly 4 people will fail this polygraph test.[2]

(c) Determine the probability that fewer than 7 people will pass this polygraph test.[3]

George goes fishing. From experience he knows that the mean number of fish he catches per hour is 1.1. It is assumed that the number of fish he catches can be modelled by a Poisson distribution. On a day in which George spends 8 hours fishing, find the probability that he will catch more than 9 fish.[4]

The weights of apples from Tony's farm follow a normal distribution with mean 158 g and standard deviation 13 g. The apples are sold in bags that contain six apples.

(a) Find the mean weight of a bag of apples.[2]

(b) Find the standard deviation of the weights of these bags of apples.[2]

(c) Find the probability that a bag selected at random weighs more than 1 kg.[2]

A company that owns many restaurants wants to determine if there are differences in the quality of the food cooked for three different meals: breakfast, lunch and dinner.

Their quality assurance team randomly selects 500 items of food to inspect. The quality of this food is classified as perfect, satisfactory, or poor. The data is summarized in the following table.

A $\chi^2$ test at the 5% significance level is carried out to determine if there is significant evidence of a difference in the quality of the food cooked for the three meals.

The critical value for this test is 9.488.

The hypotheses for this test are:

$H_0$: The quality of the food and the type of meal are independent.

$H_1$: The quality of the food and the type of meal are not independent.

(a) Find the $\chi^2$ statistic.[2]

(b) State, with justification, the conclusion for this test.[2]