Statistics and probability: 129 IB Maths AA SL practice questions

Consider the following set of ordered data.

(a) Write down[3]

(i) the mode;

(ii) the range;

(iii) the median.

(b) Find the interquartile range.[3]

Consider the following set of ordered data.

(a) Write down[3]

(i) the mode;

(ii) the range;

(iii) the median.

(b) Find the interquartile range.[3]

The random variable $X$ has the following probability distribution, with $\text{P}(X > 1) = 0.5$.

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

(b) Given that $E(X) = 1.4$, find the value of $p$ and of $q$.[6]

Events $A$ and $B$ are independent with $\mathrm{P}(A \cap B) = 0.2$ and $\mathrm{P}(A' \cap B) = 0.6$.

(a) Find $\mathrm{P}(B)$.[2]

(b) Find $\mathrm{P}(A \cup B)$.[4]

The following table shows the Mathematics test scores $(x)$ and the Science test scores $(y)$ for a group of eight students.

The regression line of $y$ on $x$ for this data can be written in the form $y = ax + b$.

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

(b) Write down the value of the Pearson's product-moment correlation coefficient, $r$.[1]

(c) Use the equation of your regression line to predict the Science test score for a student who has a score of 78 on the Mathematics test. Express your answer to the nearest integer.[2]

The following table shows the number of hours of play time, $x$, and sleep time, $y$, for a group of six children, over the period of one week.

The regression line of $y$ on $x$ for this data can be written in the form $y = ax + b$.

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

(b) Use the equation of the regression line to estimate the sleep time of a child whose weekly play time is 20 hours.[2]

Consider the following bivariate data set where $p, q \in \mathbb{Z}^+$.

The regression line of $y$ on $x$ has equation $y = 2.1875x + 0.6875$.

The regression line passes through the mean point $(\bar{x}, \bar{y})$.

(a) Given that $\bar{x} = 7$, verify that $\bar{y} = 16$.[1]

(b) Given that $q - p = 3$, find the value of $p$ and the value of $q$.[4]

The following table shows the probability distribution of a discrete random variable $X$.

(a) Find the value of $k$.[3]

(b) Find $\mathrm{E}(X)$.[3]

In a study, the mobile phone usage of a random sample of ten students was examined on a particular day.

The length of time, $t$ hours, that the ten students used their phones for are listed below.

0.7, 1.2, 1.9, 4.0, 4.4, 4.5, 4.9, 5.7, 6.5, 11.7

(a)(i) median;[1]

(a)(ii) interquartile range.[2]

An outlier is a value that is less than $Q_1 - 1.5 \times \text{IQR}$ or greater than $Q_3 + 1.5 \times \text{IQR}$.

(b) Show that 11.7 is an outlier.[3]

The heights of a group of seven-year-old children are normally distributed with mean 117 cm and standard deviation 5 cm. A child is chosen at random from the group.

(a) Find the probability that this child is taller than 122.5 cm.[3]

(b) The probability that this child is shorter than $k$ cm is 0.65. Find the value of $k$.[3]

A group of 7 adult men wanted to see if there was a relationship between their Body Mass Index (BMI) and their waist size. Their waist sizes, in centimetres, were recorded and their BMI calculated. The following table shows the results.

The relationship between $x$ and $y$ can be modelled by the regression equation $y = ax + b$.

(a) (i) Write down the value of $a$ and of $b$.[2]

(a) (ii) Find the correlation coefficient.[2]

(b) Use the regression equation to estimate the BMI of an adult man whose waist size is 95 cm.[2]

A discrete random variable, $X$, has the following probability distribution:

$$\text{P}(X = x) = \frac{kx}{20} \text{ for } x \in \{3, 5, 8, 11\}$$

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

(b) Find $E(X)$.[3]

The following table shows the probability distribution of a discrete random variable $X$.

(a) Find the value of $k$.[3]

(b) Find $E(X)$.[3]

Lucy sells hot chocolate drinks at her snack bar and has noticed that she sells more hot chocolates on cooler days. On six different days, she records the maximum daily temperature, $T$, measured in degrees centigrade, and the number of hot chocolates sold, $H$. The results are shown in the following table.

The relationship between $H$ and $T$ can be modelled by the regression line with equation $H = aT + b$.

(a)(i) Find the value of $a$ and of $b$.[3]

(a)(ii) Write down the correlation coefficient.[1]

(b) Using the regression equation, estimate the number of hot chocolates that Lucy will sell on a day when the maximum temperature is $12°C$.[2]

In Lucy's music academy, eight students took their piano diploma examination and achieved scores out of 150. For her records, Lucy decided to record the average number of hours per week each student reported practising in the weeks prior to their examination. These results are summarized in the table below.

(a) Find Pearson's product-moment correlation coefficient, $r$, for these data.[2]

(b) The relationship between the variables can be modelled by the regression equation $D = ah + b$. Write down the value of $a$ and the value of $b$.[1]

(c) One of these eight students was disappointed with her result and wished she had practised more. Based on the given data, determine how her score could have been expected to alter had she practised an extra five hours per week.[2]