5.05c Hypothesis test: normal distribution for population mean

6 The continuous random variable \(Y\) has cumulative distribution function given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < a , \\ 1 - \frac { a ^ { 3 } } { y ^ { 3 } } & y \geqslant a , \end{cases}$$ where \(a\) is a positive constant. A random sample of 3 observations, \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\), is taken, and the smallest is denoted by \(S\).

1. Show that \(\mathrm { P } ( S > s ) = \left( \frac { a } { s } \right) ^ { 9 }\) and hence obtain the probability density function of \(S\).
2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator, \(T _ { 1 }\), based on \(S\). It is given that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 2 } { 9 } \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)\), is another unbiased estimator of \(a\).
3. Given that \(\operatorname { Var } ( Y ) = \frac { 3 } { 4 } a ^ { 2 }\) and \(\operatorname { Var } ( S ) = \frac { 9 } { 448 } a ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.
4. The values of \(Y\) for a particular sample are 12.8, 4.5 and 7.0. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) for this sample, and give a reason, unrelated to efficiency, why \(T _ { 1 }\) gives a better estimate of \(a\) than \(T _ { 2 }\) in this case.

7 The continuous random variable \(U\) has unknown mean \(\mu\) and known variance \(\sigma ^ { 2 }\). In order to estimate \(\mu\), two random samples, one of 4 observations of \(U\) and the other of 6 observations of \(U\), are taken. The sample means are denoted by \(\bar { U } _ { 4 }\) and \(\bar { U } _ { 6 }\) respectively. One estimator \(S\), given by \(S = \frac { 1 } { 2 } \left( \bar { U } _ { 4 } + \bar { U } _ { 6 } \right)\), is proposed.

1. Show that \(S\) is unbiased and find \(\operatorname { Var } ( S )\) in terms of \(\sigma ^ { 2 }\). A second estimator \(T\) of the form \(a \bar { U } _ { 4 } + b \bar { U } _ { 6 }\) is proposed, where \(a\) and \(b\) are chosen such that \(T\) is an unbiased estimator for \(\mu\) with the smallest possible variance.
2. Find the values of \(a\) and \(b\) and the corresponding variance of \(T\).
3. State, giving a reason, which of \(S\) and \(T\) is the better estimator.
4. Compare the efficiencies of this preferred estimator and the mean of all 10 observations.

1 A parcel is weighed, independently, on two scales. The weights are given by the random variables \(W _ { 1 }\) and \(W _ { 2 }\) which have underlying Normal distributions as follows. $$W _ { 1 } \sim \mathrm {~N} \left( \mu , \sigma _ { 1 } ^ { 2 } \right) , \quad W _ { 2 } \sim \mathrm {~N} \left( \mu , \sigma _ { 2 } ^ { 2 } \right) ,$$ where \(\mu\) is an unknown parameter and \(\sigma _ { 1 } ^ { 2 }\) and \(\sigma _ { 2 } ^ { 2 }\) are taken as known.

1. Show that the maximum likelihood estimator of \(\mu\) is $$\hat { \mu } = \frac { \sigma _ { 2 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 1 } + \frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 2 } .$$ [You may quote the probability density function of the general Normal distribution from page 9 in the MEI Examination Formulae and Tables Booklet (MF2).]
2. Show that \(\hat { \mu }\) is an unbiased estimator of \(\mu\).
3. Obtain the variance of \(\hat { \mu }\).
4. A simpler estimator \(T = \frac { 1 } { 2 } \left( W _ { 1 } + W _ { 2 } \right)\) is proposed. Write down the variance of \(T\) and hence show that the relative efficiency of \(T\) with respect to \(\hat { \mu }\) is $$y = \left( \frac { 2 \sigma _ { 1 } \sigma _ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } \right) ^ { 2 }$$
5. Show that \(y \leqslant 1\) for all values of \(\sigma _ { 1 } ^ { 2 }\) and \(\sigma _ { 2 } ^ { 2 }\). Explain why this means that \(\hat { \mu }\) is preferable to \(T\) as an estimator of \(\mu\).

3 The human resources department of a large company is investigating two methods, A and B, for training employees to carry out a certain complicated and intricate task.

1. Two separate random samples of employees who have not previously performed the task are taken. The first sample is of size 10 ; each of the employees in it is trained by method A. The second sample is of size 12; each of the employees in it is trained by method B. After completing the training, the time for each employee to carry out the task is measured, in controlled conditions. The times are as follows, in minutes.

   Employees trained by method A:	35.2	47.8	25.8	38.0	53.6	31.0	33.9
   	35.4	21.6	42.5
   Employees trained by method B:	43.0	57.5	68.6	20.9	31.4	44.9	62.8
   	27.6	41.8	46.1	39.8	61.6

   Stating appropriate assumptions concerning the underlying populations, use a \(t\) test at the \(5 \%\) significance level to examine whether either training method is better in respect of leading, on the whole, to a lower time to carry out the task.
2. A further trial of method B is carried out to see if the performance of experienced and skilled workers can be improved by re-training them. A random sample of 8 such workers is taken. The times in minutes, under controlled conditions, for each worker to carry out the task before and after re-training are as follows.

   Worker	\(W _ { 1 }\)	\(W _ { 2 }\)	\(W _ { 3 }\)	\(W _ { 4 }\)	\(W _ { 5 }\)	\(W _ { 6 }\)	\(W _ { 7 }\)	\(W _ { 8 }\)
   Time before	32.6	28.5	22.9	27.6	34.9	28.8	34.2	31.3
   Time after	26.2	24.1	19.0	28.6	29.3	20.0	36.0	19.2

   Stating an appropriate assumption, use a \(t\) test at the \(5 \%\) significance level to examine whether the re-training appears, on the whole, to lead to a lower time to carry out the task.
3. Explain how the test procedure in part (ii) is enhanced by designing it as a paired comparison.

3 An engineering company buys a certain type of component from two suppliers, A and B. It is important that, on the whole, the strengths of these components are the same from both suppliers. The company can measure the strengths in its laboratory. Random samples of seven components from supplier A and five from supplier B give the following strengths, in a convenient unit. The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.

1. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
2. Provide a two-sided 90\% confidence interval for the true mean difference.
3. Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
4. A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.

3

1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic. A machine fills salt containers that will be sold in shops. The containers are supposed to contain 750 g of salt. The machine operates in such a way that the amount of salt delivered to each container is a Normally distributed random variable with standard deviation 20 g . The machine should be calibrated in such a way that the mean amount delivered, \(\mu\), is 750 g . Each hour, a random sample of 9 containers is taken from the previous hour's output and the sample mean amount of salt is determined. If this is between 735 g and 765 g , the previous hour's output is accepted. If not, the previous hour's output is rejected and the machine is recalibrated.
2. Find the probability of rejecting the previous hour's output if the machine is properly calibrated. Comment on your result.
3. Find the probability of accepting the previous hour's output if \(\mu = 725 \mathrm {~g}\). Comment on your result.
4. Obtain an expression for the operating characteristic of this testing procedure in terms of the cumulative distribution function \(\Phi ( z )\) of the standard Normal distribution. Evaluate the operating characteristic for the following values (in g) of \(\mu\) : 720, 730, 740, 750, 760, 770, 780.

3 At a factory, two production lines are in use for making steel rods. A critical dimension is the diameter of a rod. For the first production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.2 mm . For the second production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.4 mm . It is desired to test whether the mean diameters for the two production lines, \(\mu _ { 1 }\) and \(\mu _ { 2 }\), are equal. A random sample of 8 rods is taken from the first production line and, independently, a random sample of 10 rods is taken from the second production line.

1. Find the acceptance region for the customary test based on the Normal distribution for the null hypothesis \(\mu _ { 1 } = \mu _ { 2 }\), against the alternative hypothesis \(\mu _ { 1 } \neq \mu _ { 2 }\), at the \(5 \%\) level of significance.
2. The sample means are found to be 25.8 mm and 24.4 mm respectively. What is the result of the test? Provide a two-sided \(99 \%\) confidence interval for \(\mu _ { 1 } - \mu _ { 2 }\). The production lines are modified so that the diameters may be assumed to be of equal (but unknown) variance. However, they may no longer be Normally distributed. A two-sided test of the equality of the population medians is required, at the \(5 \%\) significance level.
3. The diameters in independent random samples of sizes 6 and 8 are as follows, in mm .

   First production line	25.9	25.8	25.3	24.7	24.4	25.4
   Second production line	23.8	25.6	24.0	23.5	24.1	24.5	24.3	25.1

   Use an appropriate procedure to carry out the test.

3 At an agricultural research station, trials are being made of two fertilisers, A and B, to see whether they differ in their effects on the yield of a crop. Preliminary investigations have established that the underlying variances of the distributions of yields using the two fertilisers may be assumed equal. Scientific analysis of the fertilisers has suggested that fertiliser A may be inferior in that it leads, on the whole, to lower yield. A statistical analysis is being carried out to investigate this. The crop is grown in carefully controlled conditions in 14 experimental plots, 6 with fertiliser A and 8 with fertiliser B. The yields, in kg per plot, are as follows, arranged in ascending order for each fertiliser.

1. Carry out a Wilcoxon rank sum test at the \(5 \%\) significance level to examine appropriate hypotheses.
2. Carry out a \(t\) test at the \(5 \%\) significance level to examine appropriate hypotheses.
3. Goodness of fit tests based on more extensive data sets from other trials with these fertilisers have failed to reject hypotheses of underlying Normal distributions. Discuss the relative merits of the analyses in parts (i) and (ii).

3

1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
2. A chemical manufacturer is endeavouring to reduce the amount of a certain impurity in one of its bulk products by improving the production process. The amount of impurity is measured in a convenient unit of concentration, and this is modelled by the Normally distributed random variable \(X\). In the old production process, the mean of \(X\), denoted by \(\mu\), was 63 and the standard deviation of \(X\) was 3.7. Experimental batches of the product are to be made using the new process, and it is desired to examine the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63\) and \(\mathrm { H } _ { 1 } : \mu < 63\) for the new process. Investigation of the variability in the new process has established that the standard deviation may be assumed unchanged. The usual Normal test based on \(\bar { X }\) is to be used, where \(\bar { X }\) is the mean of \(X\) over \(n\) experimental batches (regarded as a random sample), with a critical value \(c\) such that \(\mathrm { H } _ { 0 }\) is rejected if the value of \(\bar { X }\) is less than \(c\). The following criteria are set out.
   Find \(c\) and the smallest value of \(n\) that is required. With these values, what is the power of the test if in fact \(\mu = 58.5\) ?

3 At an agricultural research station, trials are being carried out to compare a standard variety of tomato with one that has been genetically modified (GM). The trials are concerned with the mean weight of the tomatoes and also with the aesthetic appearance of the tomatoes.

1. 1. Tomatoes of the standard and GM varieties are grown under similar conditions. The tomatoes are weighed and the data are summarised as follows.

      Variety	Sample size	Sum of weights \(( \mathrm { g } )\)	Sum of squares of weights \(\left( \mathrm { g } ^ { 2 } \right)\)
      Standard	30	3218.3	349257
      GM	26	2954.1	338691

      Carry out a test, using the Normal distribution, to investigate whether there is evidence, at the 5\% level of significance, that the two varieties of tomato differ in mean weight. State one assumption required for this test to be valid.
   2. The data in part (i) could have been used to carry out a test for the equality of means based on the \(t\) distribution. State two additional assumptions required for this test to be valid. Discuss briefly which test would be preferable in this case.
2. In order to judge whether, on the whole, GM tomatoes have a better aesthetic appearance than standard tomatoes, a trial is carried out as follows. 10 of each variety are chosen and consumer panel is asked to arrange the 20 tomatoes in order according to their appearance.
   1. State two important features of the way in which this trial should be designed. Comment briefly on how reliable the evidence from the trial is likely to be.
   2. The order in which the consumer panel arranges the tomatoes is as follows. The tomato with best appearance is listed first. \(G\) and \(S\) denote GM and standard tomatoes respectively. $$\begin{array} { c c c c c c c c c c c c c c c c c c c c } G & G & G & S & G & G & G & S & G & S & S & S & G & G & S & G & S & S & S & S \end{array}$$ Carry out an appropriate test at the \(1 \%\) level of significance.

3 A large department in a university wished to compare the standards of literacy and numeracy of its students. A random sample of 24 students was taken and sub-divided, randomly, into two groups of 12 . The students in one group took a literacy assessment (scores denoted by \(x\) ); the students in the other group took a numeracy assessment (scores denoted by \(y\) ). The two assessments were designed to give the same distributions of scores when taken by random samples from the general population. The scores obtained by the students on the two assessments are shown in the table. $$\sum x = 598 \quad \sum x ^ { 2 } = 31196 \quad \sum y = 707 \quad \sum y ^ { 2 } = 43543$$

1. Carry out an appropriate \(t\) test, at the \(5 \%\) level of significance, to compare the standards of literacy and numeracy.
2. State the distributional assumptions required for the \(t\) test to be valid. Name the test that you would use if the assumptions required for the \(t\) test are thought not to hold. State the hypotheses for this new test. Explain, in general terms, which of the two tests is more powerful, and why. A statistician at the university looked at the data and commented that a paired sample design would have been better.
3. Explain how a paired sample design would be applied in this context, and how the data would be analysed. Explain also why it would be better than the design used.

6 Gordon is a cricketer. Over a long period he knows that his population mean score, in number of runs per innings, is 28 , and the population standard deviation is 12 . In a new season he adopts a different batting style and he finds that in 30 innings using this style his mean score is 28.98 .

1. Stating a necessary assumption, test at the \(5 \%\) significance level whether his population mean score has increased.
2. Explain whether it was necessary to use the Central Limit Theorem in part (i).

6 Records for a doctors' surgery over a long period suggest that the time taken for a consultation, \(T\) minutes, has a mean of 11.0. Following the introduction of new regulations, a doctor believes that the average time has changed. She finds that, with new regulations, the consultation times for a random sample of 120 patients can be summarised as $$n = 120 , \Sigma t = 1411.20 , \Sigma t ^ { 2 } = 18737.712 .$$

1. Test, at the \(10 \%\) significance level, whether the doctor's belief is correct.
2. Explain whether, in answering part (i), it was necessary to assume that the consultation times were normally distributed.

3 An athlete finds that her times for running 100 m are normally distributed. Before a period of intensive training, her mean time is 11.8 s . After the period of intensive training, five randomly selected times, in seconds, are as follows. $$\begin{array} { l l l l l } 11.70 & 11.65 & 11.80 & 11.75 & 11.60 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to investigate whether times after the training are less, on average, than times before the training.

8 A random sample of 20 plots of land, each of equal area, was used to test whether the addition of phosphorus would increase the yield of corn. 10 plots were treated with phosphorus and 10 plots were untreated. The yields of corn, in litres, on a treated plot and on an untreated plot are denoted by \(X\) and \(Y\) respectively. You are given that $$\sum x = 2112 , \quad \sum y = 2008$$ You are also given that an unbiased estimate for the variance of treated plots is 87.96 and an unbiased estimate for the variance of untreated plots is 31.96 , both correct to 4 significant figures.

1. You may assume that the population variance estimates are sufficiently similar for the assumption of common variance to be made. What other assumption needs to be made for a \(t\)-test to be valid?
2. Carry out a suitable \(t\)-test at the \(1 \%\) significance level, to test whether the use of phosphorus increases the yield of corn.

2 In a poll of people aged 18-21, 46 out of 200 randomly chosen university students agreed with a proposition. 51 out of 300 randomly chosen others who were not university students agreed with it. Test, at the \(5 \%\) significance level, whether the proportion of university students who agree with the proposition differs from the proportion of those who are not university students.

3 A tutor gave an assessment to 6 randomly chosen new eleven-year-old students. After each student had received 30 hours of tuition, they were given a second assessment. The scores are shown in the table.

1. Show that, at the \(5 \%\) significance level, there is insufficient evidence that students' scores are higher, on average, after tuition than before tuition. State a necessary assumption.
2. Disappointed by this result, the tutor looked again at the first assessment. She discovered that the first assessment was too easy, in fact being a test for ten-year-olds, not eleven-year-olds. She decided to reduce each score for the first assessment by a constant integer \(k\). Find the least value of \(k\) for which there is evidence at the \(5 \%\) significance level that the students' scores have, on average, improved.

7 A continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < a \\ 1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a \end{array} \right.$$ where \(a\) is a parameter.
Two independent observations of \(Y\) are denoted by \(Y _ { 1 }\) and \(Y _ { 2 }\). The smaller of them is denoted by S .

1. Show that \(P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }\) and hence find the probability density function of \(S\).
2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator of \(a , T _ { 1 }\) based on \(S\).
3. Construct another unbiased estimator of \(a , T _ { 2 }\), of the form \(k \left( Y _ { 1 } + Y _ { 2 } \right)\), where \(k\) is a constant to be found.
4. Without further calculation, explain how you would decide which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.

6 The continuous random variable \(Z\) has probability density function $$f ( z ) = \left\{ \begin{array} { c c } \frac { 4 z ^ { 3 } } { k ^ { 4 } } & 0 \leqslant z \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a parameter whose value is to be estimated.

1. Show that \(\frac { 5 Z } { 4 }\) is an unbiased estimator of \(k\).
2. Find the variance of \(\frac { 5 Z } { 4 }\). The parameter \(k\) can also be estimated by making observations of a random variable \(X\) which has mean \(\frac { 1 } { 2 } k\) and variance \(\frac { 1 } { 12 } k ^ { 2 }\). Let \(Y = X _ { 1 } + X _ { 2 } + X _ { 3 }\) where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\).
3. \(c Y\) is also an unbiased estimator of \(k\). Find the value of \(c\).
4. For the value of \(c\) found in part (iii), determine which of \(\frac { 5 Z } { 4 }\) and \(c Y\) is the more efficient estimator of \(k\).

2 The continuous random variable \(Y\) has the distribution \(\mathrm { N } \left( 23.0,5.0 ^ { 2 } \right)\). The mean of \(n\) observations of \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } > 23.625 ) = 0.0228\). Find the value of \(n\).

6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).

1. Given that \(n = 81\),
   1. find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   2. find the probability that the test results in a Type II error when \(\mu = 50.2\).
   3. State how the probability of this Type II error would change if \(n\) were greater than 81 .

7 A motorist records the time taken, \(T\) minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$

1. Test, at the \(5 \%\) significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
2. Explain whether it is necessary to use the Central Limit Theorem in your test.

4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.

1. Carry out the test.
2. Explain whether it is necessary to use the Central Limit Theorem in your test.

8 In a large company the time taken for an employee to carry out a certain task is a normally distributed random variable with mean 78.0 s and unknown variance. A new training scheme is introduced and after its introduction the times taken by a random sample of 120 employees are recorded. The mean time for the sample is 76.4 s and an unbiased estimate of the population variance is \(68.9 \mathrm {~s} ^ { 2 }\).

1. Test, at the \(1 \%\) significance level, whether the mean time taken for the task has changed.
2. It is required to redesign the test so that the probability of making a Type I error is less than 0.01 when the sample mean is 77.0 s . Calculate an estimate of the smallest sample size needed, and explain why your answer is only an estimate.

5 The time \(T\) seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.

1. A test is carried out, at the \(5 \%\) significance level, of whether the specification is being met, using the mean \(\bar { t }\) of a random sample of 10 times.
   1. Find the critical region for the test, in terms of \(\bar { t }\).
   2. Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.
   3. Because of system degradation and memory load, the population mean time \(\mu\) seconds increases with the number of months of use, \(m\). A formula for \(\mu\) in terms of \(m\) is \(\mu = 20 + 0.6 m\). Use this formula to find the value of \(m\) for which the probability that the test results in rejection of the null hypothesis is 0.5 .