5.05c Hypothesis test: normal distribution for population mean

7 A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, \(x\) micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the \(10 \%\) significance level, whether the machine is producing paper of the designed thickness.

6 Records show that before the year 1990 the maximum daily temperature \(T ^ { \circ } \mathrm { C }\) at a seaside resort in August can be modelled by a distribution with mean 24.3. The maximum temperatures of a random sample of 50 August days since 1990 can be summarised by $$n = 50 , \quad \Sigma t = 1314.0 , \quad \Sigma t ^ { 2 } = 36602.17 .$$

1. Test, at the \(1 \%\) significance level, whether there is evidence of a change in the mean maximum daily temperature in August since 1990.
2. Give a reason why it is possible to use the Central Limit Theorem in your test.

5 The acidity \(A\) (measured in pH ) of soil of a particular type has a normal distribution. The pH values of a random sample of 80 soil samples from a certain region can be summarised as $$\Sigma a = 496 , \quad \Sigma a ^ { 2 } = 3126 .$$ Test, at the \(10 \%\) significance level, whether in this region the mean pH of soil is 6.1 .

8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .

1. Find the critical region for the test.
2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the \(1 \%\) significance level whether there is evidence of an increase in the mean number of floods per year.

6 The random variable \(X\) denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that \(\mathrm { E } ( X ) = 38.4\). Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the \(1 \%\) level whether there has been a change in the mean crop yield.

7 Past experience shows that \(35 \%\) of the senior pupils in a large school know the regulations about bringing cars to school. The head teacher addresses this subject in an assembly, and afterwards a random sample of 120 senior pupils is selected. In this sample it is found that 50 of these pupils know the regulations. Use a suitable approximation to test, at the \(10 \%\) significance level, whether there is evidence that the proportion of senior pupils who know the regulations has increased. Justify your approximation.

8 The random variable \(R\) has the distribution \(\mathrm { B } ( 14 , p )\). A test is carried out at the \(\alpha \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\), against \(\mathrm { H } _ { 1 } : p > 0.25\).

1. Given that \(\alpha\) is as close to 5 as possible, find the probability of a Type II error when the true value of \(p\) is 0.4 .
2. State what happens to the probability of a Type II error as
   1. \(p\) increases from 0.4,
   2. \(\alpha\) increases, giving a reason.

4 A gardening research organisation is running a trial to examine the growth and the size of flowers of various plants.

1. In the trial, seeds of three types of plant are sown. The growth of each plant is classified as good, average or poor. The results are shown in the table.

   \multirow{2}{*}{}		Growth			\multirow[t]{2}{*}{Row totals}
   		Good	Average	Poor
   \multirow{3}{*}{Type of plant}	Coriander	12	28	15	55
   	Aster	7	18	23	48
   	Fennel	14	22	11	47
   Column totals		33	68	49	150

   Carry out a test at the \(5 \%\) significance level to examine whether there is any association between growth and type of plant. State carefully your null and alternative hypotheses. Include a table of the contributions of each cell to the test statistic.
2. It is known that the diameter of marigold flowers is Normally distributed with mean 47 mm and standard deviation 8.5 mm . A certain fertiliser is expected to cause flowers to have a larger mean diameter, but without affecting the standard deviation. A large number of marigolds are grown using this fertiliser. The diameters of a random sample of 50 of the flowers are measured and the mean diameter is found to be 49.2 mm . Carry out a hypothesis test at the \(1 \%\) significance level to check whether flowers grown with this fertiliser appear to be larger on average. Use hypotheses \(\mathrm { H } _ { 0 } : \mu = 47 , \mathrm { H } _ { 1 } : \mu > 47\), where \(\mu \mathrm { mm }\) represents the mean diameter of all marigold flowers grown with this fertiliser.

3 In an English language test for 12-year-old children, the raw scores, \(X\), are Normally distributed with mean 45.3 and standard deviation 11.5.

1. Find
   (A) \(\mathrm { P } ( X < 50 )\),
   (B) \(\mathrm { P } ( 45.3 < X < 50 )\).
2. Find the least raw score which would be obtained by the highest scoring \(10 \%\) of children.
3. The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
4. In a Mathematics test for 12-year-old children, the raw scores, \(Y\), are Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\mathrm { P } ( Y < 15 ) = 0.3\) and \(\mathrm { P } ( Y < 22 ) = 0.8\), find the values of \(\mu\) and \(\sigma\).

3 The random variable \(X\) represents the reaction times, in milliseconds, of men in a driving simulator. \(X\) is Normally distributed with mean 355 and standard deviation 52.

1. Find
   (A) \(\mathrm { P } ( X < 325 )\),
   (B) \(\mathrm { P } ( 300 < X < 400 )\).
2. Find the value of \(k\) for which \(\mathrm { P } ( X < k ) = 0.2\). It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator is 344 milliseconds. You may assume that women's reaction times are also Normally distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
3. Carry out the test at the \(5 \%\) significance level.

3 The lifetime of a particular type of light bulb is \(X\) hours, where \(X\) is Normally distributed with mean 1100 and variance 2000.

1. Find \(\mathrm { P } ( 1100 < X < 1200 )\).
2. Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
3. A factory has a large number of these light bulbs installed. As soon as \(1 \%\) of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
4. The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the \(10 \%\) level to investigate the claim.
   [0pt] [Question 4 is printed overleaf.]

4

1. As part of an investigation into smoking, a random sample of 120 students was selected. The students were asked whether they were smokers, and also whether either of their parents were smokers. The results are summarised in the table below. Test, at the \(5 \%\) significance level, whether there is any association between the smoking habits of the students and their parents.

   	At least one parent smokes	Neither parent smokes
   Student smokes	21	27
   Student does not smoke	17	55

2. The manufacturer of a particular brand of cigarette claims that the nicotine content of these cigarettes is Normally distributed with mean 0.87 mg . A researcher suspects that the mean nicotine content of this brand is higher than the value claimed by the manufacturer. The nicotine content, \(x \mathrm { mg }\), is measured for a random sample of 100 cigarettes. The data are summarised as follows. $$\sum x = 88.20 \quad \sum x ^ { 2 } = 78.68$$ Carry out a test at the \(1 \%\) significance level to investigate the researcher's belief. \section*{END OF QUESTION PAPER}

4

1. A random sample of 80 GCSE students was selected to take part in an investigation into whether attitudes to mathematics differ between girls and boys. The students were asked if they agreed with the statement 'Mathematics is one of my favourite subjects'. They were given three options 'Agree', 'Disagree', 'Neither agree nor disagree'. The results, classified according to sex, are summarised in the table below.

   	Agree	Disagree	Neither
   Male	17	13	8
   Female	12	11	19

   The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.

   	Agree	Disagree	Neither
   Male	0.7550	0.2246	1.8153
   Female	0.6831	0.2032	1.6424

   1. Calculate the expected frequency for females who agree. Verify the corresponding contribution, 0.6831 , to the test statistic.
   2. Carry out the test at the \(5 \%\) level of significance.
2. The level of radioactivity in limpets (a type of shellfish) in the sea near to a nuclear power station is regularly monitored. Over a period of years it has been found that the level (measured in suitable units) is Normally distributed with mean 5.64. Following an incident at the power station, a researcher suspects that the mean level of radioactivity in limpets may have increased. The researcher selects a random sample of 60 limpets. Their levels of radioactivity, \(x\) (measured in the same units), are summarised as follows. $$\sum x = 373 \quad \sum x ^ { 2 } = 2498$$ Carry out a test at the \(5 \%\) significance level to investigate the researcher's belief.

5 The concentration level of mercury in a large lake is known to have a normal distribution with standard deviation 0.24 in suitable units. At the beginning of June 2008, the mercury level was measured at five randomly chosen places on the lake, and the sample mean is denoted by \(\bar { x } _ { 1 }\). Towards the end of June 2008 there was a spillage in the lake which may have caused the mercury level to rise. Because of this the level was then measured at six randomly chosen points of the lake, and the mean of this sample is denoted by \(\bar { x } _ { 2 }\).

1. State hypotheses for a test based on the two samples for whether, on average, the level of mercury had increased. Define any parameters that you use.
2. Find the set of values of \(\bar { x } _ { 2 } - \bar { x } _ { 1 }\) for which there would be evidence at the 5\% significance level that, on average, the level of mercury had increased.
3. Given that the average level had actually increased by 0.3 units, find the probability of making a Type II error in your test, and comment on its value.

7 A tutor gives a randomly selected group of 8 students an English Literature test, and after a term's further teaching, she gives the group a similar test. The marks for the two tests are given in the table.

1. Stating a necessary condition, show by carrying out a suitable \(t\)-test, at the \(1 \%\) significance level, that the marks do not give evidence of an improvement.
2. The tutor later found that she had marked the second test too severely, and she decided to add a constant amount \(k\) to each mark. Find the least integer value of \(k\) for which the increased marks would give evidence of improvement at the \(1 \%\) significance level.

3 It is given that \(X _ { 1 }\) and \(X _ { 2 }\) are independent random variables with \(X _ { 1 } \sim \mathrm {~N} \left( \mu _ { 1 } , 2.47 \right)\) and \(X _ { 2 } \sim \mathrm {~N} \left( \mu _ { 2 } , 4.23 \right)\). Random samples of \(n _ { 1 }\) observations of \(X _ { 1 }\) and \(n _ { 2 }\) observations of \(X _ { 2 }\) are taken. The sample means are denoted by \(\bar { X } _ { 1 }\) and \(\bar { X } _ { 2 }\).

1. State the distribution of \(\bar { X } _ { 1 } - \bar { X } _ { 2 }\), giving its parameters. For two particular samples, \(n _ { 1 } = 5 , \Sigma x _ { 1 } = 48.25 , n _ { 2 } = 10\) and \(\Sigma x _ { 2 } = 72.30\).
2. Test at the \(2 \%\) significance level whether \(\mu _ { 1 }\) differs from \(\mu _ { 2 }\). A student stated that because of the Central Limit Theorem the sample means will have normal distributions so it is unnecessary for \(X _ { 1 }\) and \(X _ { 2 }\) to have normal distributions.
3. Comment on the student's statement.

5 Each of a random sample of 200 steel bars taken from a production line was examined and 27 were found to be faulty.

1. Find an approximate \(90 \%\) confidence interval for the proportion of faulty bars produced. A change in the production method was introduced which, it was claimed, would reduce the proportion of faulty bars. After the change, each of a further random sample of 100 bars was examined and 8 were found to be faulty.
2. Test the claim, at the \(10 \%\) significance level.

6 The deterioration of a certain drug over time was investigated as follows. The drug strength was measured in each of a random sample of 8 bottles containing the drug. These were stored for two years and the strengths were then re-measured. The original and final strengths, in suitable units, are shown in the following table.

1. Stating any required assumption, test at the \(5 \%\) significance level whether the mean strength has decreased by more than 0.2 over the two years.
2. Calculate a 95\% confidence interval for the mean reduction in strength over the two years.

2 A new running track has been developed and part of the testing procedure involves 7 randomly chosen athletes. They each run 100 m on both the old and new tracks.
The results are as follows. The population mean times on the old and new tracks are denoted by \(\mu _ { \mathrm { O } }\) seconds and \(\mu _ { \mathrm { N } }\) seconds respectively. Stating any necessary assumption, carry out a suitable \(t\)-test of the null hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } = 0\) against the alternative hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } > 0\). Use a \(2 \frac { 1 } { 2 } \%\) significance level .

3 Two reading schemes, \(A\) and \(B\), are compared by using them with a random sample of 9 five-year-old children. The children are divided into two groups, 5 allotted to scheme \(A\) and 4 to scheme \(B\), and the schemes are taught under similar conditions.
After one year the children are given the same test and their scores, \(x _ { A }\) and \(x _ { B }\), are summarised below. With the usual notation, $$\begin{aligned} & n _ { A } = 5 , \bar { x } _ { A } = 52.0 , \sum \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 248 , \\ & n _ { B } = 4 , \bar { x } _ { B } = 56.5 , \sum \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 381 . \end{aligned}$$ It may be assumed that scores have normal distributions.

1. Calculate an \(80 \%\) confidence interval for the difference in population mean scores for the two methods.
2. State a further assumption required for the validity of the interval.

8 After contracting a particular disease, patients from a hospital are advised to have their blood tested monthly for a year. In order to test whether patients comply with this advice the hospital management commissioned a survey of 100 patients. A hospital statistician selected the patients randomly from records and asked the patients whether or not they had complied with the advice. The results classified by gender are as follows.

1. Test at the \(5 \%\) significance level whether compliance with the advice is independent of gender.
2. A manager believed that a greater proportion of female patients than male patients comply with the advice. Carry out an appropriate test of proportions at the \(10 \%\) significance level.

3 A machine produces circular metal discs whose radii have a normal distribution with mean \(\mu \mathrm { cm }\). A random sample of five discs is selected and their radii, in cm, are as follows. $$\begin{array} { l l l l l } 6.47 & 6.52 & 6.46 & 6.47 & 6.51 \end{array}$$

1. Calculate a \(95 \%\) confidence interval for \(\mu\).
2. Hence state a 95\% confidence interval for the mean circumference of a disc.

4 In order to compare the difficulty of two Su Doku puzzles, two random samples of 40 fans were selected. One sample was given Puzzle 1 and the other sample was given Puzzle 2. Of those given Puzzle 1, 24 could solve it within ten minutes. Of those given Puzzle 2, 15 could solve it within ten minutes.

1. Using proportions, test at the \(5 \%\) significance level whether there is a difference in the standard of difficulty of the two puzzles.
2. The setter believed that Puzzle 2 was more difficult than Puzzle 1. Obtain the smallest significance level at which this belief is supported.

5 Each person in a random sample of 15 men and 17 women from a university campus was asked how many days in a month they took exercise. The numbers of days for men and women, \(x _ { M }\) and \(x _ { W }\) respectively, are summarised by $$\Sigma x _ { M } = 221 , \quad \Sigma x _ { M } ^ { 2 } = 3992 , \quad \Sigma x _ { W } = 276 , \quad \Sigma x _ { W } ^ { 2 } = 5538 .$$

1. State conditions for the validity of a suitable test of the difference in the mean numbers of days for men and women on the campus.
2. Given that these conditions hold, carry out the test at the \(5 \%\) significance level.
3. If in fact the random sample was drawn entirely from the university Mathematics Department, state with a reason whether the validity of the test is in doubt.