Date | May 2016 | Marks available | 7 | Reference code | 16M.3sp.hl.TZ0.4 |
Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
Command term | Calculate and Determine | Question number | 4 | Adapted from | N/A |
Question
The owner of a factory is asked to produce bricks of weight 2.2 kg. The quality control manager wishes to test whether or not, on a particular day, the mean weight of bricks being produced is 2.2 kg.
He therefore collects a random sample of 20 of these bricks and determines the weight, \(x\) kg, of each brick. He produces the following summary statistics.
\[\sum {x = 42.0,{\text{ }}\sum {{x^2} = 89.2} } \]
State hypotheses to enable the quality control manager to test the mean weight using a two-tailed test.
(i) Calculate unbiased estimates of the mean and the variance of the weights of the bricks being produced.
(ii) Assuming that the weights of the bricks are normally distributed, determine the \(p\)-value of the above results and state the conclusion in context using a 5% significance level.
The owner is more familiar with using confidence intervals. Determine a 95% confidence interval for the mean weight of bricks produced on that particular day.
Markscheme
\({H_0}:{\text{ }}\mu = 2.2;{\text{ }}{H_1}:{\text{ }}\mu \ne 2.2\) A1A1
[2 marks]
(i) UE of mean \( = \frac{{42.0}}{{20}}{\text{ = }}2.1\) A1
UE of variance \( = \frac{{89.2}}{{19}} - \frac{{20 \times {{2.1}^2}}}{{19}} = 0.0526{\text{ }}\left( {\frac{1}{{19}}} \right)\) (M1)A1
Note: Award (M0) for division by 20 where there is no subsequent use of \(\frac{{20}}{{19}}\).
(ii)
\(t = - 1.95\) (A1)
\({\text{DF}} = 19\) (A1)
\(p - value = 0.0662\) A1
Note: Allow follow through from (b)(i). In particular, 0.05 for the variance gives \(t = - 2\) and \(p\)-value 0.0600.
accept \({H_0}\), or equivalent statement involving \({H_0}\) or \({H_1}\), indicating that the mean weight is 2.2kg R1
Note: Follow through the candidate’s \(p\)-value.
[7 marks]
\([1.99,{\text{ }}2.21]\) A1A1
Note: Allow follow through from (b)(i). In particular, 0.05 for the variance gives \([2.00,{\text{ }}2.20]\).
[2 marks]
Examiners report
Most candidates stated the correct hypotheses in (a).
In (b)(i), the mean was invariably found correctly, although to find the variance estimate, quite a few candidates divided by 20 instead of 19. Incorrect variances were followed through in the next part of (b)(i). The \(t\)-test was generally well applied and the correct conclusion drawn. It was, however, surprising to note that many candidate used the appropriate formula to find the value of \(t\) and hence the \(p\)-value as opposed to using their GDC software.
Part (c) was generally well answered.
Syllabus sections
- 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...
- 16M.3sp.hl.TZ0.2b: (i) Determine the \(p\)-value. (ii) State your conclusion at the 5% significance level.
- 16M.3sp.hl.TZ0.2c: Explain why the equation of the regression line of \(y\) on \(x\) should not be used to...
- 16M.3sp.hl.TZ0.3a: Find the value of \(k\).
- 16M.3sp.hl.TZ0.3b: (i) Calculate an unbiased estimate for \(\theta \), using the random sample, 8.3, 4.2,...
- 16M.3sp.hl.TZ0.3c: (i) Show that \({\text{Var}}(U) = \frac{{{\theta ^2}}}{{6n}}\). (ii) Show that...
- 16M.3sp.hl.TZ0.4a: State hypotheses to enable the quality control manager to test the mean weight using a...
- 16M.3sp.hl.TZ0.4c: The owner is more familiar with using confidence intervals. Determine a 95% confidence...
- 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...
- 16M.3sp.hl.TZ0.1a: Given that, on a randomly chosen day, the probability that he completes the crossword in less...
- 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...
- 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...
- 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...
- 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...
- 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....
- 15N.3sp.hl.TZ0.2c: If the moisture content of a beam is found to be \(9.5\), use the appropriate regression line...
- 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...
- 15N.3sp.hl.TZ0.1a: One hundred men from that country, selected at random, had their heights measured. The mean...
- 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...
- 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...
- 12N.3sp.hl.TZ0.3b: The random variable Y represents the height of a wave on another surf beach. It is known that...
- 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...
- 12N.3sp.hl.TZ0.4e: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided...
- 12N.3sp.hl.TZ0.4f: Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided...
- 08M.3sp.hl.TZ1.1: A coin was tossed 200 times and 115 of these tosses resulted in ‘heads’. Use a two-tailed...
- 08M.3sp.hl.TZ1.3: (a) Find the probability that the weight of a randomly chosen apple is more than double...
- 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 %...
- 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...
- 08N.3sp.hl.TZ0.2: The apple trees in a large orchard have, for several years, suffered from a disease for which...
- 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...
- 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...
- 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...
- 09M.3sp.hl.TZ0.1: Ahmed and Brian live in the same house. Ahmed always walks to school and Brian always cycles...
- 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...
- 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...
- 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...
- SPNone.3sp.hl.TZ0.3a: Find the probability that the weight of a randomly selected male is more than twice the...
- SPNone.3sp.hl.TZ0.3b: Two males and five females stand together on a weighing machine. Find the probability that...
- 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....
- 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...
- 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...
- 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....
- 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...
- 13M.3sp.hl.TZ0.5a: Find the probability that (i) he hits the target exactly 4 times in his first 8...
- 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...
- 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...
- 13M.3sp.hl.TZ0.5b: Ben hits the target for the \({10^{{\text{th}}}}\) time with his \({X^{{\text{th}}}}\)...
- 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...
- 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...
- 11M.2.hl.TZ1.1a: Estimate the (i) median weight of the apples; (ii) 30th percentile of the weight of...
- 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) ...
- 14M.3sp.hl.TZ0.2: The following table gives the average yield of olives per tree, in kg, and the rainfall, in...
- 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...
- 13N.3sp.hl.TZ0.1c: Let \(J\) be the 90% confidence interval for the mean speed. Without calculating \(J\),...
- 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...
- 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.
- 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...
- 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...
- 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...
- 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...
- 15M.3sp.hl.TZ0.3c: Find the confidence level of the interval that corresponds to halving the width of...
- 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...
- 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...
- 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...
- 14N.3sp.hl.TZ0.4e: By consideration of the probability generating function, \({G_{X + Y}}(t)\), of \(X + Y\),...
- 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...
- 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...
- 14N.3sp.hl.TZ0.5d: If, having done the test, the sample mean is found to lie within the critical region, find...