DP Mathematics HL Questionbank

Description

The aims of this option are to allow students the opportunity to approach statistics in a practical way; to demonstrate a good level of statistical understanding; and to understand which situations apply and to interpret the given results. It is expected that GDCs will be used throughout this option, and that the minimum requirement of a GDC will be to find probability distribution function (pdf), cumulative distribution function (cdf), inverse cumulative distribution function, p-values and test statistics, including calculations for the following distributions: binomial, Poisson, normal and t. Students are expected to set up the problem mathematically and then read the answers from the GDC, indicating this within their written answers. Calculator-specific or brand-specific language should not be used within these explanations.

Directly related questions

• 18M.3sp.hl.TZ0.4c.i: Show that Cov(U, V) = E(UV) − E(U)E(V).
• 18M.3sp.hl.TZ0.4b.ii: State your conclusion at the 1 % significance level.
• 18M.3sp.hl.TZ0.4b.i: Determine the p-value.
• 18M.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not a negative linear association exists...
• 18M.3sp.hl.TZ0.3d: Another model of smartphone whose battery life may be assumed to be normally distributed with...
• 18M.3sp.hl.TZ0.3c: Find the probability of making a Type II error.
• 18M.3sp.hl.TZ0.3b: Find the critical region for testing ${\bar b}$ at the 5 % significance level.
• 18M.3sp.hl.TZ0.3a: State suitable hypotheses for a two-tailed test.
• 18M.3sp.hl.TZ0.2d: Determine the mean number of throws required to obtain a 1.
• 18M.3sp.hl.TZ0.2c: Find $G'\left( t \right)$.
• 18M.3sp.hl.TZ0.2b: Show that the probability generating function, $G\left( t \right)$, for X is given by...
• 18M.3sp.hl.TZ0.2a: State the distribution of X.
• 18M.3sp.hl.TZ0.1c: Two randomly chosen male birds and three randomly chosen female birds are placed on a weighing...
• 18M.3sp.hl.TZ0.1b: Find the probability that the weight of a randomly chosen male bird is more than twice the weight...
• 18M.3sp.hl.TZ0.1a: Find the probability that a randomly chosen male bird weighs between 4.75 kg and 4.85 kg.
• 16M.3sp.hl.TZ0.4c: The owner is more familiar with using confidence intervals. Determine a 95% confidence interval...
• 16M.3sp.hl.TZ0.4b: (i)     Calculate unbiased estimates of the mean and the variance of the weights of the bricks...
• 16M.3sp.hl.TZ0.4a: State hypotheses to enable the quality control manager to test the mean weight using a two-tailed...
• 16M.3sp.hl.TZ0.3c: (i)     Show that ${\text{Var}}(U) = \frac{{{\theta ^2}}}{{6n}}$. (ii)     Show that ${U^2}$...
• 16M.3sp.hl.TZ0.3b: (i)     Calculate an unbiased estimate for $\theta$, using the random sample, 8.3, 4.2, 6.5,...
• 16M.3sp.hl.TZ0.2c: Explain why the equation of the regression line of $y$ on $x$ should not be used to predict...
• 16M.3sp.hl.TZ0.2b: (i)     Determine the $p$-value. (ii)     State your conclusion at the 5% significance level.
• 16M.3sp.hl.TZ0.2a: State suitable hypotheses to investigate whether or not $X$, $Y$ are independent.
• 16M.3sp.hl.TZ0.1a: Given that, on a randomly chosen day, the probability that he completes the crossword in less...
• 16N.3sp.hl.TZ0.4b: By differentiating $J(t)$, prove that (i)    ...
• 16N.3sp.hl.TZ0.4a: Write down an expression for $J(t)$ in terms of $G(t)$ and $H(t)$.
• 16N.3sp.hl.TZ0.3b: (i)     Calculate ${\text{E}}(Y)$. (ii)     Calculate ${\text{P}}(Y = 5)$.
• 16N.3sp.hl.TZ0.3a: (i)     State the distribution of $X$, including its parameters. (ii)     Calculate...
• 16N.3sp.hl.TZ0.2c: Use this model to find the values of $A$ and $B$ such that...
• 16N.3sp.hl.TZ0.2b: Explain why a normal distribution can be used to give an approximate model for $X$.
• 16N.3sp.hl.TZ0.2a: Find (i)     ${\text{E}}(X)$; (ii)     ${\text{Var}}(X)$.
• 16N.3sp.hl.TZ0.2d: Calculate the probability that he makes a Type II error.
• 16N.3sp.hl.TZ0.1d: (i)     State the distribution of your test statistic (including any parameters). (ii)     Find...
• 16N.3sp.hl.TZ0.1c: Using a suitable regression line, find an estimate for his score on Exam 2, giving your answer to...
• 16N.3sp.hl.TZ0.1b: (i)     State the distribution of the test statistic (including any parameters). (ii)     Find...
• 16N.3sp.hl.TZ0.1a: For these data find the product moment correlation coefficient, $r$.
• 16M.3sp.hl.TZ0.1b: Find the probability that the total time taken for him to complete five randomly chosen...
• 16M.3sp.hl.TZ0.1c: Find the probability that, on a randomly chosen day, the time taken by Beatrice to complete the...
• 16M.3sp.hl.TZ0.3a: Find the value of $k$.
• 16M.3sp.hl.TZ0.5a: Show that the probability distribution of $Y$ is given by...
• 16M.3sp.hl.TZ0.5b: (i)     Show that $G(t)$, the probability generating function of $Y$, is given by...
• 17N.3sp.hl.TZ0.5c.ii: Identify the probability distribution given in part (c)(i) and state its parameters.
• 17N.3sp.hl.TZ0.5c.i: Show...
• 17N.3sp.hl.TZ0.5b: By considering the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, show that...
• 17N.3sp.hl.TZ0.5a.ii: Hence show that ${\text{Var}}(X) = \lambda$.
• 17N.3sp.hl.TZ0.5a.i: Find expressions for ${G’_X}(t)$ and ${G’’_X}(t)$.
• 17N.3sp.hl.TZ0.4b: Find the least value of $|r|$ for which the test concludes that $\rho  \ne 0$.
• 17N.3sp.hl.TZ0.4a: State suitable hypotheses to investigate whether or not $U$, $V$ are independent.
• 17N.3sp.hl.TZ0.3b.iii: Hence find an expression for the most efficient estimator and interpret the result.
• 17N.3sp.hl.TZ0.3b.ii: Find, in terms of ${n_1}$ and ${n_2}$, an expression for $a$ which gives the most efficient...
• 17N.3sp.hl.TZ0.3b.i: Show that...
• 17N.3sp.hl.TZ0.3a: Show that $U = a{\bar X_1} + (1 - a){\bar X_2},{\text{ }}a \in \mathbb{R}$, is an unbiased...
• 17N.3sp.hl.TZ0.2b.ii: Interpret your $p$-value at the 5% level of significance, justifying your conclusion.
• 17N.3sp.hl.TZ0.2a: Determine unbiased estimates for $\mu$ and ${\sigma ^2}$.
• 17N.3sp.hl.TZ0.2b.i: Use a two-tailed test to determine the $p$-value for the above results.
• 17N.3sp.hl.TZ0.1b.ii: Given that $P(T < a) = 0.75$, find the value of $a$.
• 17N.3sp.hl.TZ0.1b.i: Sketch the graph of $F(t)$ for $0 \leqslant t \leqslant 2$, clearly indicating the...
• 17N.3sp.hl.TZ0.1a: Find the cumulative distribution function $F(t)$, for $0 \leqslant t \leqslant 2$.
• 17N.3dm.hl.TZ0.2a: Find an expression for ${u_n}$ in terms of $n$.
• 17M.3sp.hl.TZ0.5c: The teacher then asks Anne for the values of the $t$-statistic and the product moment...
• 17M.3sp.hl.TZ0.5b: State, in context, what conclusion should be drawn from this $p$-value.
• 17M.3sp.hl.TZ0.5a: State suitable hypotheses for this investigation.
• 17M.3sp.hl.TZ0.4c.ii: Hence find the variance of this best unbiased estimator.
• 17M.3sp.hl.TZ0.4c.i: Hence find the value of $a$ and the value of $b$ which give the best unbiased estimator of...
• 17M.3sp.hl.TZ0.4b: Show that ${\text{Var}}(U) = (39{b^2} - 12b + 1){\sigma ^2}$.
• 17M.3sp.hl.TZ0.4a: Given that $U$ is unbiased, show that $2a + 6b = 1$.
• 17M.3sp.hl.TZ0.3c: The random variable $Y$ is given by $Y = 2X + 1$. Find the probability generating function...
• 17M.3sp.hl.TZ0.3b: Hence determine ${\text{E}}(X)$ in terms of $p$ and $q$.
• 17M.3sp.hl.TZ0.3a: Show that the probability generating function for $X$ is given by...
• 17M.3sp.hl.TZ0.2c.ii: A random sample of 100 observations is obtained from the distribution of $X$. If $\bar X$...
• 17M.3sp.hl.TZ0.2c.i: State the central limit theorem. 
• 17M.3sp.hl.TZ0.2b.ii: Hence determine the mean and the variance of $X$.
• 17M.3sp.hl.TZ0.2b.i: Show that the probability density function $f$ of $X$ is given, for...
• 17M.3sp.hl.TZ0.2a.ii: Determine the median of $X$.
• 17M.3sp.hl.TZ0.2a.i: Determine $P(0.25 \leqslant X \leqslant 0.75)$;
• 17M.3sp.hl.TZ0.1c.ii: Using a 10% significance level and justifying your answer, state your conclusion in context.
• 17M.3sp.hl.TZ0.1c.i: Carry out an appropriate test and state the $p$-value obtained.
• 17M.3sp.hl.TZ0.1b: Find unbiased estimates of $\mu$ and ${\sigma ^2}$.
• 17M.3sp.hl.TZ0.1a: State suitable hypotheses to test the inspector’s claim.
• 15N.3sp.hl.TZ0.5b: The die is rolled a second time, and the score ${X_2}$ is noted. (i)     Show that...
• 15N.3sp.hl.TZ0.5a: (i)     Find ${\text{E}}({X_1})$. (ii)     Hence obtain an unbiased estimator for $p$.
• 15N.3sp.hl.TZ0.4f: A third random variable $W$, has probability generating function...
• 15N.3sp.hl.TZ0.4e: A third random variable $W$, has probability generating function...
• 15N.3sp.hl.TZ0.4c: Prove that the probability generating function of $U$ is given by...
• 15N.3sp.hl.TZ0.4b: Hence, or otherwise, find the value of $P(U > 20)$.
• 15N.3sp.hl.TZ0.4a: Find $F(u)$, the cumulative distribution function of $U$, for...
• 15N.3sp.hl.TZ0.3: Two students are selected at random from a large school with equal numbers of boys and girls. The...
• 15N.3sp.hl.TZ0.2c: If the moisture content of a beam is found to be $9.5$, use the appropriate regression line to...
• 15N.3sp.hl.TZ0.2a: Determine the product moment correlation coefficient for these data.
• 15N.3sp.hl.TZ0.1b: A second random sample of size $n$ is taken from the same population. Find the minimum value of...
• 15N.3sp.hl.TZ0.1a: One hundred men from that country, selected at random, had their heights measured. The mean of...
• 12M.3sp.hl.TZ0.1b: In spite of these results the baker insists that his claim is correct. Stating appropriate...
• 12M.3sp.hl.TZ0.2c: Given that p = 0.2, find the least value of n for which...
• 12M.3sp.hl.TZ0.1a: Determine unbiased estimates for the mean and variance of the distribution.
• 12M.3sp.hl.TZ0.2a: Show that ${\text{P}}(X \leqslant n) = 1 - {(1 - p)^n},{\text{ }}n \in {\mathbb{Z}^ + }$ .
• 12M.3sp.hl.TZ0.2b: Deduce an expression for...
• 12M.3sp.hl.TZ0.4c: Two independent observations are made from X and the values are added. The resulting random...
• 12M.3sp.hl.TZ0.4d: (i)     Find the cumulative distribution function for X . (ii)     Hence, or otherwise, find the...
• 12N.3sp.hl.TZ0.2a: Find the mean and the variance of (i)     ${X_1} + {X_2}$ ; (ii)     $3{X_1}$; (iii)    ...
• 12N.3sp.hl.TZ0.2b: Find ${\text{E}}(X_1^2)$ in terms of $\mu$ and $\sigma$ .
• 12N.3sp.hl.TZ0.3a: The random variable X represents the height of a wave on a particular surf beach. It is known...
• 12N.3sp.hl.TZ0.3b: The random variable Y represents the height of a wave on another surf beach. It is known that Y...
• 12N.3sp.hl.TZ0.4a: If the dice is fair, write down the distribution of X , including the value of any parameter(s).
• 12N.3sp.hl.TZ0.4b: Write down E(X ) for the distribution in part (a).
• 12N.3sp.hl.TZ0.4d: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice....
• 12N.3sp.hl.TZ0.4e: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice....
• 12N.3sp.hl.TZ0.4f: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice....
• 08M.3sp.hl.TZ1.1: A coin was tossed 200 times and 115 of these tosses resulted in ‘heads’. Use a two-tailed test...
• 08M.3sp.hl.TZ1.3: (a)     Find the probability that the weight of a randomly chosen apple is more than double the...
• 08M.3sp.hl.TZ1.4: (a)     Calculate an unbiased estimate for   (i)     $\mu$ ,   (ii)     ${\sigma ^2}$...
• 08M.3sp.hl.TZ1.5: (a)     Having thrown the die once, she lets ${X_2}$ denote the number of additional throws...
• 08M.3sp.hl.TZ2.1a: The random variable Y is such that \({\text{E}}(2Y + 3) = 6{\text{ and Var}}(2 - 3Y) =...
• 08M.3sp.hl.TZ2.1b: Independent random variables R and S are such...
• 08M.3sp.hl.TZ2.2: A factory makes wine glasses. The manager claims that on average 2 % of the glasses are...
• 08M.3sp.hl.TZ2.3: (a)     State appropriate null and alternative hypotheses. (b)     Test at the 5 % significance...
• 08M.3sp.hl.TZ2.5: (a)     Find the appropriate critical regions corresponding to a significance level of   (i)    ...
• 08N.3sp.hl.TZ0.3: (a)     The heating in a residential school is to be increased on the third frosty day during the...
• 08N.3sp.hl.TZ0.2: The apple trees in a large orchard have, for several years, suffered from a disease for which the...
• 08N.3sp.hl.TZ0.4: (a)     A random variable, X , has probability density function defined...
• 11M.3sp.hl.TZ0.1b: Five of these oranges are selected at random to be put into a bag. Find the probability that the...
• 11M.3sp.hl.TZ0.3a: Determine unbiased estimates of the mean and variance of the loss in weight achieved over the...
• 11M.3sp.hl.TZ0.4b: Given that the value of $\mu$ is actually 2.5, determine the probability of a Type II error.
• 11M.3sp.hl.TZ0.5b: Jo has a biased coin which has a probability of 0.35 of showing heads when tossed. She tosses...
• 11M.3sp.hl.TZ0.1c: The farm also produces lemons whose weights may be assumed to be normally distributed with mean...
• 11M.3sp.hl.TZ0.3b: (i)     State suitable hypotheses for testing whether or not this diet causes a mean loss in...
• 11M.3sp.hl.TZ0.4a: Calculate the significance level of the test.
• 11M.3sp.hl.TZ0.5a: The random variable X has the negative binomial distribution NB(3, p) . Let $f(x)$ denote the...
• 09M.3sp.hl.TZ0.1: Ahmed and Brian live in the same house. Ahmed always walks to school and Brian always cycles to...
• 09M.3sp.hl.TZ0.2: (a)     After a chemical spillage at sea, a scientist measures the amount, x units, of the...
• 09M.3sp.hl.TZ0.4: In a game there are n players, where $n > 2$ . Each player has a disc, one side of which is...
• 09N.3sp.hl.TZ0.2a: Alan and Brian are athletes specializing in the long jump. When Alan jumps, the length of his...
• 09N.3sp.hl.TZ0.1: The mean weight of a certain breed of bird is believed to be 2.5 kg. In order to test this...
• 09N.3sp.hl.TZ0.2b: Colin joins the squad and the coach wants to know the mean length, $\mu$ metres, of his jumps....
• SPNone.2.hl.TZ0.3a: Complete the grouped frequency table for these data.  
• SPNone.2.hl.TZ0.3b: Estimate the mean and standard deviation of the heights of these 80 boys.
• SPNone.2.hl.TZ0.3c: Explain briefly whether or not the normal distribution provides a suitable model for this...
• SPNone.3sp.hl.TZ0.1a: Determine unbiased estimates of $\mu$ and ${\sigma ^2}$.
• SPNone.3sp.hl.TZ0.1c: The stallholder claims that the mean weight of apples is 125 grams but the shopper claims that...
• SPNone.3sp.hl.TZ0.3a: Find the probability that the weight of a randomly selected male is more than twice the weight of...
• SPNone.3sp.hl.TZ0.3b: Two males and five females stand together on a weighing machine. Find the probability that their...
• SPNone.3sp.hl.TZ0.1b: Determine a 99 % confidence interval for $\mu$ .
• SPNone.3sp.hl.TZ0.2b: Show that the probability generating function for X is given...
• SPNone.3sp.hl.TZ0.2c: Hence determine ${\text{E}}(X)$.
• SPNone.3sp.hl.TZ0.4a: State suitable hypotheses.
• SPNone.3sp.hl.TZ0.4b: The marks obtained by the 12 students who sat both papers are given in the following...
• SPNone.3sp.hl.TZ0.4c: George obtained a mark of 63 on Paper 1 but was unable to sit Paper 2 because of illness. Predict...
• SPNone.3sp.hl.TZ0.4d: Another class of 16 students sat examinations in Physics and Chemistry and the product moment...
• SPNone.3sp.hl.TZ0.5b: In order to estimate $\theta$, a random sample of n observations is obtained from the...
• 10M.3sp.hl.TZ0.1: Anna cycles to her new school. She records the times taken for the first ten days with the...
• 10M.3sp.hl.TZ0.2: The random variable X has a Poisson distribution with mean $\mu$. The value of $\mu$ is...
• 10M.3sp.hl.TZ0.4: A shop sells apples, pears and peaches. The weights, in grams, of these three types of fruit may...
• 10M.3sp.hl.TZ0.5b: The random variable X has the negative binomial distribution NB(5, p), where p < 0.5, and...
• 10N.2.hl.TZ0.2: The company Fresh Water produces one-litre bottles of mineral water. The company wants to...
• 10N.3sp.hl.TZ0.1a: A hospital specializes in treating overweight patients. These patients have weights that are...
• 10N.3sp.hl.TZ0.2: The length of time, T, in months, that a football manager stays in his job before he is removed...
• 10N.3sp.hl.TZ0.3: As soon as Sarah misses a total of 4 lessons at her school an email is sent to her parents. The...
• 10N.3sp.hl.TZ0.4: A teacher has forgotten his computer password. He knows that it is either six of the letter J...
• 13M.3sp.hl.TZ0.1a: Find unbiased estimates of $\mu$ and ${\sigma ^2}$.
• 13M.3sp.hl.TZ0.4b: (i)     State the central limit theorem. (ii)     A random sample of 150 observations is taken...
• 13M.3sp.hl.TZ0.5a: Find the probability that (i)     he hits the target exactly 4 times in his first 8 shots; (ii)...
• 13M.3sp.hl.TZ0.1b: Determine a 95 % confidence interval for $\mu$.
• 13M.3sp.hl.TZ0.1c: Given the hypotheses $${{\text{H}}_0}:\mu  = 15;{\text{ }}{{\text{H}}_1}:\mu  \ne 15,$$ find...
• 13M.3sp.hl.TZ0.3a: State suitable hypotheses for this investigation.
• 13M.3sp.hl.TZ0.3b: It is decided to define the critical region by $x \leqslant 25$. (i)     Calculate the...
• 13M.3sp.hl.TZ0.4a: (i)     Determine an expression for $F(x)$, valid for $1 \leqslant x \leqslant 2$, where F...
• 13M.3sp.hl.TZ0.5b: Ben hits the target for the ${10^{{\text{th}}}}$ time with his ${X^{{\text{th}}}}$ shot. (i)...
• 11N.3sp.hl.TZ0.3d: Find the median of X in terms of $\lambda$.
• 11N.3sp.hl.TZ0.1c: Find the probability that five randomly chosen Supermug tea bags contain a total of less than...
• 11N.3sp.hl.TZ0.1d: Find the probability that the total weight of tea in seven randomly chosen Supermug tea bags is...
• 11N.3sp.hl.TZ0.3b: Find the cumulative distribution function, $F(x)$, of X.
• 11N.3sp.hl.TZ0.3c: Find the probability that the lifetime of a particular battery is more than twice the mean.
• 11N.3sp.hl.TZ0.3e: Find the probability that the lifetime of a particular battery lies between the median and the mean.
• 11M.2.hl.TZ1.1a: Estimate the (i)     median weight of the apples; (ii)     30th percentile of the weight of the...
• 11M.2.hl.TZ1.1b: Estimate the number of apples which weigh more than 110 grams.
• 14M.3sp.hl.TZ0.1b: $\bar X$ denotes the sample mean of $n > 1$ independent observations from $X$. (i)    ...
• 14M.3sp.hl.TZ0.1c: A random sample of $40$ observations is taken from the distribution for $X$. (i)     Find...
• 14M.3sp.hl.TZ0.2: The following table gives the average yield of olives per tree, in kg, and the rainfall, in cm,...
• 14M.3sp.hl.TZ0.3: (a)     Consider the random variable $X$ for which ${\text{E}}(X) = a\lambda  + b$, where...
• 14M.3sp.hl.TZ0.4: Consider the random variable $X \sim {\text{Geo}}(p)$. (a)     State \({\text{P}}(X <...
• 13N.3sp.hl.TZ0.1c: Let $J$ be the 90% confidence interval for the mean speed. Without calculating $J$, explain...
• 13N.3sp.hl.TZ0.3c: (i)     State the meaning of a type II error. (ii)     Write down how to proceed if it is...
• 13N.3sp.hl.TZ0.4: Francisco and his friends want to test whether performance in running 400 metres improves if they...
• 13N.3sp.hl.TZ0.5: Let $X$ and $Y$ be independent random variables with $X \sim {P_o}{\text{ (3)}}$ and...
• 13N.3sp.hl.TZ0.1b: Find the 95% confidence interval, $I$, for the mean speed.
• 13N.3sp.hl.TZ0.3b: Explain what can be done with this data to decrease the probability of making a type I error.
• 18M.3sp.hl.TZ0.5b.ii: Hence write down an unbiased estimator of Var(X).
• 18M.3sp.hl.TZ0.5b.i: Show that ${\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)$.
• 18M.3sp.hl.TZ0.5a: Show that $P = \frac{X}{n}$ is an unbiased estimator of $p$.
• 18M.3sp.hl.TZ0.4c.ii: Hence show that if U, V are independent random variables then the population product moment...
• 15M.3sp.hl.TZ0.2a: Determine unbiased estimates of $\mu$ and ${\sigma ^2}$.
• 15M.3sp.hl.TZ0.3b: Interpret the result found in (a).
• 15M.3sp.hl.TZ0.4a: Explain briefly the meaning of (i)     an estimator of $\mu$; (ii)     an unbiased estimator...
• 15M.3sp.hl.TZ0.1b: A large can and a small can are selected at random. Find the probability that the large can...
• 15M.3sp.hl.TZ0.4b: A random sample ${X_1},{\text{ }}{X_2},{\text{ }}{X_3}$ of three independent observations is...
• 15M.3sp.hl.TZ0.5c: Two independent random variables ${X_1}$ and ${X_2}$ are such that...
• 15M.3sp.hl.TZ0.1c: A large can and five small cans are selected at random. Find the probability that the large can...
• 15M.3sp.hl.TZ0.2b: (i)     State suitable hypotheses to test the claim that extra tuition improves examination...
• 15M.3sp.hl.TZ0.3a: Calculate a $99\%$ confidence interval for $\mu$. Give your answer correct to three decimal...
• 15M.3sp.hl.TZ0.3c: Find the confidence level of the interval that corresponds to halving the width of the $99\%$...
• 15M.3sp.hl.TZ0.5a: Determine the probability generating function for $X \sim {\text{B}}(1,{\text{ }}p)$.
• 15M.3sp.hl.TZ0.5b: Explain why the probability generating function for ${\text{B}}(n,{\text{ }}p)$ is a polynomial...
• 14N.3sp.hl.TZ0.1c: Find the interquartile range for $X$.
• 14N.3sp.hl.TZ0.2b: (i)     Find the expected number of throws required for Eric to hit the target three times. (ii)...
• 14N.3sp.hl.TZ0.4a: (i)     Prove that ${\text{E}}(X) = \lambda$. (ii)     Prove that...
• 14N.3sp.hl.TZ0.4b: $Y$ is a random variable, independent of $X$, that also follows a Poisson distribution with...
• 14N.3sp.hl.TZ0.4e: By consideration of the probability generating function, ${G_{X + Y}}(t)$, of $X + Y$, prove...
• 14N.3sp.hl.TZ0.4f: Find (i)     ${G_{X + Y}}(1)$; (ii)     ${G_{X + Y}}( - 1)$.
• 14N.3sp.hl.TZ0.5a: Find the critical region for this test.
• 14N.3sp.hl.TZ0.2c: If he has just $$8$, find the probability he will lose all his money before he hits the target...
• 14N.3sp.hl.TZ0.3a: If $X$ and $Y$ are two random variables such that ${\text{E}}(X) = {\mu _X}$ and...
• 14N.3sp.hl.TZ0.4g: Hence find the probability that $X + Y$ is an even number.
• 14N.3sp.hl.TZ0.1b: Find the cumulative distribution function for $X$.
• 14N.3sp.hl.TZ0.3b: In a particular company, it is claimed that the distance travelled by employees to work is...
• 14N.3sp.hl.TZ0.4c: Let $T = \frac{Y}{2} + \frac{Y}{2}$. (i)     Show that $T$ is an unbiased estimator for...
• 14N.3sp.hl.TZ0.5c: It is now known that in the area in which the plant was found $90\%$ of all the plants are of...
• 14N.3sp.hl.TZ0.5d: If, having done the test, the sample mean is found to lie within the critical region, find the...