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...