One-sample t-test, variance unknown

1. A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70 . To test this claim a car magazine measures the number of miles per gallon, \(x\), of each of a random sample of 20 Panther cars and obtained the following statistics.

$$\bar { x } = 71.2 \quad s = 3.4$$ The number of miles per gallon may be assumed to be normally distributed.

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the manufacturer's claim. The standard deviation of the number of miles per gallon for the Tiger car is 4 .
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the variance of the number of miles per gallon for the Panther car is different from that of the Tiger car.

5. Boxes of chocolates manufactured by Philippe have a mean weight of \(\mu\) grams and a standard deviation of \(\sigma\) grams. A random sample of 25 of these boxes are weighed. Using this sample, the unbiased estimate of \(\mu\) is 455 and the unbiased estimate of \(\sigma ^ { 2 }\) is 55 .

1. Test, at the \(5 \%\) level of significance, whether or not \(\sigma\) is greater than 6 . State your hypotheses clearly.
2. Test, at the \(5 \%\) level of significance, whether or not \(\mu\) is more than 450 .
3. State an assumption you have made in order to carry out the above tests.

1. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml, in each bottle is recorded. Unbiased estimates for the mean and variance are

$$\hat { \mu } = 502 \quad s ^ { 2 } = 5.6$$ Stating your hypotheses clearly, test at the 1\% level of significance

1. whether or not the mean amount of water in a bottle is more than 500 ml ,
2. whether or not the standard deviation of the amount of water in a bottle is less than 3 ml .

1. George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.

$$\sum x = 113.4 \quad \sum x ^ { 2 } = 1414.08$$ The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not these data support George’s belief.

1. At the start of each academic year, a large college carries out a diagnostic test on a random sample of new students. Past experience has shown that the standard deviation of the scores on this test is 19.71

The admissions tutor claimed that the new students in 2013 would have more varied scores than usual. The scores for the students taking the test can be assumed to come from a normal distribution. A random sample of 10 new students was taken and the score \(x\), for each student was recorded. The data are summarised as \(\sum x = 619 \sum x ^ { 2 } = 42397\)

1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test the admission tutor's claim. The admissions tutor decides that in future he will use the same hypotheses but take a larger sample of size 30 and use a significance level of 1\%.
2. Use the tables to show that, to 3 decimal places, the critical region for \(S ^ { 2 }\) is \(S ^ { 2 } > 664.281\)
3. Find the probability of a type II error using this test when the true value of the standard deviation is in fact 22.20

1. A production line is designed to fill bottles with oil. The amount of oil placed in a bottle is normally distributed and the mean is set to 100 ml .

The amount of oil, \(x \mathrm { ml }\), in each of 8 randomly selected bottles is recorded, and the following statistics are obtained. $$\bar { x } = 92.875 \quad s = 8.3055$$ Malcolm believes that the mean amount of oil placed in a bottle is less than 100 ml .
Stating your hypotheses clearly, test, at the \(5 \%\) significance level, whether or not Malcolm's belief is supported.

2. The weights of piglets at birth, \(M \mathrm {~kg}\), are normally distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) A random sample of 9 piglets is taken and their weights at birth, \(m \mathrm {~kg}\), are recorded. The results are summarised as $$\sum m = 11.6 \quad \sum m ^ { 2 } = 15.2$$ Stating your hypotheses clearly, test at the 5\% level of significance

1. whether or not the mean weight of piglets at birth is greater than 1.2 kg ,
2. whether or not the standard deviation of the weights of piglets at birth is different from 0.3 kg .

3. The lengths, \(X \mathrm {~mm}\), of the wings of adult blackbirds follow a normal distribution. A random sample of 5 adult blackbirds is taken and the lengths of the wings are measured. The results are summarised below $$\sum x = 655 \text { and } \sum x ^ { 2 } = 85845$$

1. Test, at the \(10 \%\) level of significance, whether or not the mean length of an adult blackbird's wing is less than 135 mm . State your hypotheses clearly.
2. Find the \(90 \%\) confidence interval for the variance of the lengths of adult blackbirds' wings. Show your working clearly.

1. A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) are

$$\bar { x } = 202 \quad \text { and } \quad s ^ { 2 } = 3.6$$ Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean weight of almonds in a packet is more than 200 g .

1. The weights of the contents of jars of jam are normally distributed with a stated mean of 100 g . A random sample of 7 jars was taken and the contents of each jar, \(x\) grams, was weighed. The results are summarised by the following statistics.

$$\sum x = 710.9 , \sum x ^ { 2 } = 72219.45 .$$ Test at the \(5 \%\) level of significance whether or not there is evidence that the mean weight of the contents of the jars is greater than 100 g . State your hypotheses clearly.
(8 marks)

5 A technician is investigating whether a batch of nylon 66 (a particular type of nylon) is contaminated by another type of nylon.
The average melting point of nylon 66 is \(264 ^ { \circ } \mathrm { C }\). However, if the batch is contaminated by the other type of nylon the melting point will be lower. The melting points, in \({ } ^ { \circ } \mathrm { C }\), of a random sample of 8 pieces of nylon from the batch are as follows. \(\begin{array} { l l l l l l l l } 262.7 & 265.0 & 264.1 & 261.7 & 262.9 & 263.5 & 261.3 & 262.6 \end{array}\)

1. Find
   • the sample mean,
   • the sample standard deviation.
   The technician produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in Fig. 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1db026a2-dffc-4877-b927-247fbf0e7a78-5_560_1358_982_246} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure}
2. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data.
3. In this question you must show detailed reasoning. Carry out a suitable test at the \(5 \%\) significance level to investigate whether the batch appears to be contaminated with another type of nylon.
4. Name an alternative test that could have been carried out if the population standard deviation had been known.

11 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows. $$\begin{array} { l l l l l l l l l l } - 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14 \end{array}$$ A Normal probability plot and the \(p\)-value of the Kolmogorov-Smirnov test for these data are shown below. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}

1. The researcher decides to carry out a hypothesis test in order to investigate the claim. Comment on the type of hypothesis test that should be used. You should refer to
   • The Normal probability plot
   • The \(p\)-value of the Kolmogorov-Smirnov test
   • Carry out a test at the \(5 \%\) significance level to investigate whether the researcher's belief may be correct.
   • If the Normal probability plot had been different, giving a \(p\)-value of 0.65 for the KolmogorovSmirnov test, a different procedure could have been used to investigate the researcher's belief.
   • State what alternative test could have been used in this case.
   • State what the hypotheses would have been.

7 An environmental investigator wants to check whether the level of selenium in carrots in fields near a mine is different from the usual level in the country, which is \(9.4 \mathrm { ng } / \mathrm { g }\) (nanograms per gram). She takes a random sample of 10 carrots from fields near the mine and measures the selenium level of each of them in \(\mathrm { ng } / \mathrm { g }\), with results as follows. \(\begin{array} { l l l l l l l l l l } 6.20 & 10.72 & 11.42 & 16.32 & 15.33 & 10.56 & 8.83 & 9.21 & 7.78 & 14.32 \end{array}\)

1. Find estimates of each of the following.
   • The population mean
   • The population standard deviation
   The investigator produces a Normal probability plot and carries out a Kolmogorov-Smirnov test for these data as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bab116b3-6e5f-44db-ac86-670e4040d649-06_583_1499_959_242}
2. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data.
3. State the null hypothesis for the Kolmogorov-Smirnov test for Normality.
4. In this question you must show detailed reasoning. Carry out a test at the \(5 \%\) significance level to investigate whether the mean selenium level in carrots from fields near the mine is different from \(9.4 \mathrm { ng } / \mathrm { g }\).
5. If the \(p\)-value of the Kolmogorov-Smirnov test for Normality had been 0.007, explain what procedure you could have used to investigate the selenium level in carrots from fields near the mine.

1. The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.
   305
   295
   310
   310
   315
   307
   300
   311
   306

You may assume the times taken to boil the full kettle are normally distributed.

1. Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.
2. Test, at the \(5 \%\) level of significance, whether there is evidence to suggest that this is an old kettle.
3. State a factor that Alun should control when carrying out this investigation.

7 Petroxide Industries produces a chemical used in the production of mobile phone covers for a mobile phone company. The chemical becomes less effective when the mean level of impurity is greater than 3 per cent.
Sunita is the Quality Control manager at Petroxide Industries. After a complaint from the mobile phone company, Sunita obtains a random sample of this chemical from 9 batches. She measures the level of impurity, \(X\) per cent, in each sample.
The summarised results are as follows. $$\sum x = 28.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 0.6$$ 7

1. 1. Investigate using the \(5 \%\) level of significance whether the mean level of impurity in the chemical is greater than 3 per cent.
      [0pt] [7 marks]
      7
2. (ii) State the assumption that it was necessary for you to make in order for the test in part (a)(i) to be valid.
   7
3. State the changes that would be required to your test in part (a) if you were told that the standard deviation of the level of impurity is known to be 0.25 per cent.
   [0pt] [2 marks]
   Turn over for the next question

6 A company manufactures bolts. The diameter of the bolts follows a normal distribution with a mean diameter of 5 mm . Stan believes that the mean diameter of the bolts is less than 5 mm . He takes a random sample of 10 bolts and measures their diameters. He calculates some statistics but spills ink on his work before completing them. The only information he has left is as follows \includegraphics[max width=\textwidth, alt={}, center]{67df73d4-6ce4-45f7-8a69-aa94292ea814-16_394_1150_527_456} Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not Stan's belief is supported.

8 A jam producer claims that the mean weight of jam in a jar is 230 grams.

1. A random sample of 8 jars is selected and the weight of jam in each jar is determined. The results, in grams, are $$\begin{array} { l l l l l l l l } 220 & 228 & 232 & 219 & 221 & 223 & 230 & 229 \end{array}$$ Assuming that the weight of jam in a jar is normally distributed, test, at the \(5 \%\) level of significance, the jam producer's claim.
2. It is later discovered that the mean weight of jam in a jar is indeed 230 grams. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out the hypothesis test in part (a). Give a reason for your answer.

4. A recent census in the U.K. revealed that the heights of females in the U.K. have a mean of 160.9 cm . A doctor is studying the heights of female Indians in a remote region of South America. The doctor measured the height, \(x \mathrm {~cm}\), of each of a random sample of 30 female Indians and obtained the following statistics. $$\Sigma x = 4400.7 , \quad \Sigma \mathrm { x } ^ { 2 } = 646904.41 .$$ The heights of female Indians may be assumed to follow a normal distribution.
The doctor presented the results of the study in a medical journal and wrote 'the female Indians in this region are more than 10 cm shorter than females in the U.K.'

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the doctor's statement.
   (6) The census also revealed that the standard deviation of the heights of U.K. females was 6.0 cm .
2. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the variance of the heights of female Indians is different from that of females in the U.K.
   (6)

7 A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text { and } \quad \sum x ^ { 2 } = 10198$$ Investigate the shopkeeper's claim using the \(5 \%\) level of significance.
State any assumptions that you make.

7 The rainfall per day in February in a particular town has been recorded as having a mean of 1.8 inches. Sienna claims that rainfall in February has increased in the town. She records the rainfall in a random sample of 12 days. Her sample mean is 2 inches and her sample standard deviation is 0.4 inches.
It is assumed that rainfall per day has a normal distribution.
7

1. Investigate Sienna's claim using the \(5 \%\) level of significance.
   7
2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
3. The distribution of rainfall per day in February in the town over 10 years is shown in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{443e7f17-a555-41ff-9d91-541cf45aae99-11_508_645_849_699} Explain whether or not the assumption that rainfall per day in February has a normal distribution is appropriate.

4 The random variable \(X\) has a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\) A random sample of 8 observations of \(X\) has mean \(\bar { x } = 101.5\) and gives the unbiased estimate of the variance as \(s ^ { 2 } = 4.8\) The random sample is used to conduct a hypothesis test at the \(10 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 100 \\ & \mathrm { H } _ { 1 } : \mu \neq 100 \end{aligned}$$ Carry out the hypothesis test.

6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.