DP Further Mathematics HL Questionbank

Description

The aims of this topic 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 the 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.2.hl.TZ0.8d: Let \(M = \frac{1}{2}N\). Show that M has a geometric distribution and hence find the value of E(N).
• 18M.2.hl.TZ0.8c: After 6 serves the score is 3 points each. Play continues and the game ends when one player has...
• 18M.2.hl.TZ0.8b: Two friends A and B play a ball game with the following rules. Each player starts with zero...
• 18M.2.hl.TZ0.8a.ii: Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).
• 18M.2.hl.TZ0.8a.i: Show that the probability generating function of X is given...
• 18M.2.hl.TZ0.1c: Given that \({\bar X}\) and \({\bar Y}\) are the respective sample means,...
• 18M.2.hl.TZ0.1b.iii: P(X1 + X2 + Y1 + Y2 + Y3 + Y4 < 30).
• 18M.2.hl.TZ0.1b.ii: P(3X1 + 4Y1 > 15).
• 18M.2.hl.TZ0.1b.i: P(X1 + Y1 < 11).
• 18M.2.hl.TZ0.1a: Write down the distribution of aX + bY where a, b \( \in \mathbb{R}\).
• 18M.1.hl.TZ0.7b.ii: Find the critical value for \({\bar x}\) if she wants the probabilities of a Type I error and a...
• 18M.1.hl.TZ0.7b.i: She decides to use the same acceptance criteria as the previous investigator. Find the...
• 18M.1.hl.TZ0.7a: An investigator wishes to test the hypotheses H0 : μ = 65, H1 : μ > 65. He decides on the...
• 18M.1.hl.TZ0.14b: The null hypothesis μ = 46.5 is tested against the alternative hypothesis μ < 46.5 at the λ%...
• 18M.1.hl.TZ0.14a: Calculate a 90% confidence interval for the population mean mark μ for this paper.
• 16M.2.hl.TZ0.2d: Find the probability that the sample mean is less than 2.3.
• 16M.2.hl.TZ0.2c: State the central limit theorem.
• 16M.2.hl.TZ0.2b: (i)     Find \({\text{E}}({X^2})\). (ii)     Show that \({\text{Var}}(X) = 2\).
• 16M.1.hl.TZ0.4d: State the conclusions that the president of the club should reach from this test, giving reasons...
• 16M.1.hl.TZ0.4c: (i)     Find unbiased estimates of \(\mu \) and \({\sigma ^2}\). (ii)     Find the value of the...
• 16M.1.hl.TZ0.4b: (i)     Give a reason why a \(t\) test is appropriate and write down its degrees of...
• 16M.1.hl.TZ0.4a: State suitable hypotheses.
• 16M.1.hl.TZ0.2c: Find the probability that the total lifetime of 5 randomly chosen robust vacuum cleaners is...
• 16M.1.hl.TZ0.2b: Find the probability that the total lifetime of 7 randomly chosen basic vacuum cleaners is less...
• 16M.1.hl.TZ0.2a: Find \({\text{P}}\left( {B > {\text{E}}(B) + \frac{1}{2}\sqrt {{\text{Var}}(B)} } \right)\).
• 16M.1.hl.TZ0.13c: Find the set of values of \(n\) for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).
• 16M.1.hl.TZ0.13b: Find \({\text{E}}({X_n})\).
• 16M.1.hl.TZ0.13a: Use the formula for the sum of a finite geometric series to show...
• 17M.2.hl.TZ0.7b.iii: Hence find \({\text{Var}}(X)\).
• 17M.2.hl.TZ0.7b.ii: By differentiating both sides of this equation, determine the values of \(G’(1)\) and \(G’’(1)\).
• 17M.2.hl.TZ0.7b.i: Show that \(\ln G(t) = \ln 4 + \ln t - 2\ln (3 - t)\).
• 17M.2.hl.TZ0.7a.iv: Hence show that \(k = \frac{4}{3}\).
• 17M.2.hl.TZ0.7a.iii: By considering \(\left( {1 - \frac{t}{3}} \right)G(t)\), show...
• 17M.2.hl.TZ0.7a.ii: Determine the radius of convergence of this infinite series.
• 17M.2.hl.TZ0.7a.i: Write down the first three terms of the infinite series for \(G(t)\), the probability generating...
• 17M.1.hl.TZ0.4b: Determine the probability that their combined weight exceeds the recommended maximum.
• 17M.1.hl.TZ0.4a: Find the probability that the weight of a randomly chosen male student is more than twice the...
• 17M.1.hl.TZ0.13b.ii: Give a correct interpretation.
• 17M.1.hl.TZ0.13b.i: Explain briefly why this is an incorrect statement.
• 17M.1.hl.TZ0.13a.ii: Hence show that, with probability...
• 17M.1.hl.TZ0.13a.i: State the distribution of \(\frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}\).
• 17M.1.hl.TZ0.1b: Calculate unbiased estimates of the mean and the variance of the weights of this breed of bird.
• 17M.1.hl.TZ0.1a: State suitable hypotheses for testing the above claim using a two-tailed test.
• 15M.2.hl.TZ0.6b: Show that...
• 15M.2.hl.TZ0.6a: Find \({\text{E}}(A)\).
• 15M.2.hl.TZ0.2c: Test, at the \(1\%\) level of significance, the null hypothesis \(\mu  = 35\) against the...
• 15M.2.hl.TZ0.2b: The sample values are summarized as \(\sum {x = 3782} \) and \(\sum {{x^2} = 155341} \) where...
• 15M.2.hl.TZ0.2a: State the distribution of \(\bar X\) , giving its mean and standard deviation.
• 15M.1.hl.TZ0.14: Sarah is the quality control manager for the Stronger Steel Corporation which makes steel sheets....
• 15M.1.hl.TZ0.7b: Sami now considers the brand “Bright”. The weight of the contents of a randomly chosen packet of...
• 15M.1.hl.TZ0.7a: Sami is undertaking market research on packets of soap powder. He considers the brand “Gleam”....
• 15M.1.hl.TZ0.5e: Estimate the foot length of a boy of height 170 cm.
• 15M.1.hl.TZ0.5d: Find the equation of the regression line of \(y\) on \(x\).
• 15M.1.hl.TZ0.5c: Interpret the \(p\)-value in the context of the question.
• 15M.1.hl.TZ0.5b: Find the \(p\)-value.
• 15M.1.hl.TZ0.5a: Calculate the product moment correlation coefficient.
• 11M.1.hl.TZ0.1a: Bottles of iced tea are supposed to contain 500 ml. A random sample of 8 bottles was selected and...
• 11M.1.hl.TZ0.1b: A random sample of size four is taken from the distribution N(60, 36) . Calculate the...
• 10M.2.hl.TZ0.4a: The weights, \(X\) grams, of tomatoes may be assumed to be normally distributed with mean...
• 10M.2.hl.TZ0.4b: The random variable \(Y\) has variance \({\sigma ^2}\) , where \({\sigma ^2} > 0\) . A random...
• 09M.2.hl.TZ0.1A.b: (i)      Use your answer to (a) to find an approximate expression for the cumulative distributive...
• 09M.2.hl.TZ0.1B.b: What is the critical region for the sample mean if the probability of a Type I error is to be...
• 09M.2.hl.TZ0.1B.c: If the mean weight of the bags is actually \(28\).1 kg, what would be the probability of a Type...
• 09M.2.hl.TZ0.1B.a: State and justify an appropriate test procedure giving the null and alternate hypotheses.
• 13M.1.hl.TZ0.5a: (i)     Find \({\rm{E}}(Y)\) in the form \(p\mu \) , where \(p \in \mathbb{R}\) . (ii)     Find...
• 13M.2.hl.TZ0.3a: (i)     Find an expression for \({\rm{P}}(X > a)\) , where \(a > 0\) . A chicken crosses a...
• 13M.2.hl.TZ0.3b: A rifleman shoots at a circular target. The distance in centimetres from the centre of the target...
• 13M.1.hl.TZ0.5b: A random sample of \(n\) values of \(Y\) was found to have a mean of \(8.76\).   (i)     Given...
• 13M.2.hl.TZ0.1a: (i)     Write down the mode of \(X\) . (ii)     Find the exact value of \(p\) if...
• 13M.2.hl.TZ0.1b: (i)     Find the smallest value of \(n\) for which the probability of Arthur walking to school on...
• 08M.2.hl.TZ0.2B.a: He makes 5 independent measurements of the concentration of a particular solution and correctly...
• 08M.2.hl.TZ0.2B.b: He is now given a different solution and is asked to determine a \(95\%\) confidence interval for...
• 07M.1.hl.TZ0.6a: (i)     Find the mean and variance of \(2Y - X\) . (ii)     Find the probability that the weight...
• 07M.1.hl.TZ0.6b: Two randomly chosen male birds and three randomly chosen female birds are placed together on a...
• 07M.2.hl.TZ0.2b: An automatic machine is used to fill bottles of water. The amount delivered, \(Y\) ml , may be...
• 12M.1.hl.TZ0.5a: The shopkeeper claims that when one of the coins is tossed, the probability of obtaining a head...
• 12M.1.hl.TZ0.5b: Bill tosses the other coin a large number of times and counts the number of heads obtained. He...
• 12M.2.hl.TZ0.5a: The continuous random variable \(X\) takes values only in the interval [\(a\), \(b\)] and \(F\)...
• SPNone.1.hl.TZ0.4a: The shopkeeper places \(100\) randomly chosen potatoes on a weighing machine. Find the...
• SPNone.1.hl.TZ0.4b: Find the minimum number of randomly selected potatoes which are needed to ensure that their total...
• 12M.2.hl.TZ0.5b: The continuous random variable \(Y\) has probability density function \(f\) given...
• SPNone.1.hl.TZ0.10a: (i)     Calculate the correlation coefficient for this sample. (ii)     Calculate the...
• SPNone.1.hl.TZ0.10b: (i)     Calculate the equation of the least squares regression line of \(w\) on \(h\) . (ii)    ...
• SPNone.2.hl.TZ0.2a: The farm manager selects a random sample of \(10\) apples and weighs them with the following...
• SPNone.2.hl.TZ0.2b: The farm manager claims that the mean weight of apples is \(100\) grams but the buyer from the...
• SPNone.2.hl.TZ0.7a: (i)     Find an expression for \(c\) in terms of \(n\) such that...
• SPNone.2.hl.TZ0.7b: (i)     Show that \({\rm{P}}(Y \le y) = {\left( {\frac{y}{a}} \right)^{3n}},0 \le y \le a\) and...
• SPNone.2.hl.TZ0.7c: Show that \(\frac{{{\rm{Var}}(U)}}{{{\rm{Var}}(V)}} = \frac{{3n + 2}}{5}\) and hence state, with...
• 14M.1.hl.TZ0.3: The following table shows the probability distribution of the discrete random variable...
• 14M.1.hl.TZ0.7: The weights, in grams, of 10 apples were measured with the following results:      \(212.2\)    ...
• 14M.1.hl.TZ0.11: The random variables \(X\), \(Y\) follow a bivariate normal distribution with product moment...
• 14M.2.hl.TZ0.1: The random variable \(X\) has the binomial distribution \({\text{B}}(n,{\text{ }}p)\), where...
• 15M.2.hl.TZ0.6c: Find \({\text{P}}(A \leqslant 5|A > 3)\).