5.05c Hypothesis test: normal distribution for population mean

1. Fred is a new employee in a delicatessen. He is asked to cut cheese into 100 g blocks. A random sample of 8 of these blocks of cheese is selected. The weight, in grams, of each block of cheese is given below

$$94 , \quad 106 , \quad 115 , \quad 98 , \quad 111 , \quad 104 , \quad 113 , \quad 102$$

1. Calculate a \(90 \%\) confidence interval for the standard deviation of the weights of the blocks of cheese cut by Fred. Given that the weights of the blocks of cheese are independent,
2. state what further assumption is necessary for this confidence interval to be valid. The delicatessen manager expects the standard deviation of the weights of the blocks of cheese cut by an employee to be less than 5 g. Any employee who does not achieve this target is given training.
3. Use your answer from part (a) to comment on Fred's results. A second employee, Olga, has just been given training. Olga is asked to cut cheese into 100 g blocks. A random sample of 20 of these blocks of cheese is selected. The weight of each block of cheese, \(x\) grams, is recorded and the results are summarised below. $$\bar { x } = 102.6 \quad s ^ { 2 } = 19.4$$ Given that the assumption in part (b) is also valid in this case,
4. test, at a \(10 \%\) level of significance, whether or not the mean weight of the blocks of cheese cut by Olga after her training is 100 g . State your hypotheses clearly.
   (6)

1. As part of their research two sports science students, Ali and Bea, select a random sample of 10 adult male swimmers and a random sample of 13 adult male athletes from local sports clubs. They measure the arm span, \(x \mathrm {~cm}\), of each person selected.
   The data are summarised in the table below

The students know that the arm spans of adult male swimmers and of adult male athletes may each be assumed to be normally distributed.
They decide to share out the data analysis, with Ali investigating the means of the two distributions and Bea investigating the variances of the two distributions. Ali assumes that the variances of the two distributions are equal. She calculates the pooled estimate of variance, \(s _ { p } { } ^ { 2 }\)

1. Show that \(s _ { p } { } ^ { 2 } = 112.6\) to 1 decimal place. Ali claims that there is no difference in the mean arm spans of adult male swimmers and of adult male athletes.
2. Stating your hypotheses clearly, test this claim at the \(10 \%\) level of significance.
   (5) Bea believes that the variances of the arm spans of adult male swimmers and adult male athletes are not equal.
3. Show that, at the \(10 \%\) level of significance, the data support Bea's belief. State your hypotheses and show your working clearly. Ali and Bea combine their work and present their results to their tutor, Clive.
4. Explain why Clive is not happy with their research and state, with a reason, which of the tests in parts (b) and (c) is not valid.
   

1. A researcher is investigating the accuracy of IQ tests. One company offers IQ tests that it claims will give any individual's IQ with a standard deviation of 5

The researcher takes these tests 9 times with the following results $$123 , \quad 118 , \quad 127 , \quad 120 , \quad 134 , \quad 120 , \quad 118 , \quad 135 , \quad 121$$

1. Find the sample mean, \(\bar { x }\), and the sample variance, \(s ^ { 2 }\), of these scores.
   (2) Given that any individual's IQ scores on these tests are independent and have a normal distribution,
2. use the hypotheses $$\mathrm { H } _ { 0 } : \sigma ^ { 2 } = 25 \text { against } \mathrm { H } _ { 1 } : \sigma ^ { 2 } > 25$$ to test the company's claim at the \(5 \%\) significance level.
   (4) Gurdip works for the company and has taken these IQ tests 12 times. Gurdip claims that the sample variance of these 12 scores is \(s ^ { 2 } = 8.17\)
3. Use this value of \(s ^ { 2 }\) to calculate a \(95 \%\) confidence interval for the variance of Gurdip's IQ test scores.
   [0pt] [You may use \(\mathrm { P } \left( \chi _ { 11 } ^ { 2 } > 3.816 \right) = 0.975\) and \(\mathrm { P } \left( \chi _ { 11 } ^ { 2 } > 21.920 \right) = 0.025\) ]
4. Assuming that \(\sigma ^ { 2 } = 25\), comment on Gurdip's claim.

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 .

5. Fire brigades in cities \(X\) and \(Y\) are in similar locations. The response times, in minutes, during a particular month, for randomly selected calls are summarised in the table below. You may assume that the response times are from independent normal distributions.
Stating your hypotheses and showing your working clearly

1. test, at the \(10 \%\) level of significance, whether or not the variances of the populations from which the response times are drawn are the same,
   (5)
2. test, at the \(5 \%\) level of significance, whether or not the mean response time for the fire brigade in city \(X\) is more than 5 minutes longer than the mean response time for the fire brigade in city \(Y\).
3. Explain why your result in part (a) enables you to carry out the test in part (b).

1. The times taken by children to run 150 m are normally distributed. The times taken, \(x\) seconds, by a random sample of 9 boys and an independent random sample of 6 girls are recorded. The following statistics are obtained.

1. Test, at the \(10 \%\) level of significance, whether or not the variances of the two distributions are equal. State your hypotheses clearly. The Headteacher claims that the mean time taken for the girls is more than 5 seconds greater than the mean time taken for the boys.
2. Stating your hypotheses clearly, test the Headteacher's claim. Use a \(1 \%\) level of significance and show your working clearly.

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.

4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.

1. 1. State why a paired \(t\)-test is suitable for use with these data.
   2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
2. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
3. Explain, in the context of the coach's belief, what a Type II error would be in this case.

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. A random sample of 8 students is selected from a school database.

Each student's reaction time is measured at the start of the school day and again at the end of the school day. The reaction times, in milliseconds, are recorded below.

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
   (1) The random variable \(R\) is the reaction time at the start of the school day minus the reaction time at the end of the school day. The mean of \(R\) is \(\mu\). John uses a paired \(t\)-test to test the hypotheses $$\mathrm { H } _ { 0 } : \mu = m \quad \mathrm { H } _ { 1 } : \mu \neq m$$ Given that \(\mathrm { H } _ { 0 }\) is rejected at the 5\% level of significance but accepted at the 1\% level of significance,
2. find the ranges of possible values for \(m\).

1. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5\% significance level,

• the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail and
• the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.

A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded. It can be assumed that the force at which each glue fails is normally distributed.

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the standard deviation of the force required for Goglue to fail is greater than the standard deviation of the force required for the Tackfast to fail. State your hypotheses clearly. The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the supplier's claim.
3. Show that, at the \(5 \%\) level of significance, the supplier's claim will be accepted if \(\bar { X } _ { G } - \bar { X } _ { T } > 4.55\), where \(\bar { X } _ { G }\) and \(\bar { X } _ { T }\) are the mean forces required for Goglue to fail and Tackfast to fail respectively. Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
4. Explain whether or not this information affects the decision about which glue the supplier decides to use.

1. A machine makes posts. The length of a post is normally distributed with unknown mean \(\mu\) and standard deviation 4 cm .

A random sample of size \(n\) is taken to test, at the \(5 \%\) significance level, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 150 \quad \mathrm { H } _ { 1 } : \mu > 150$$

1. State the probability of a Type I error for this test. The manufacturer requires the probability of a Type II error to be less than 0.1 when the actual value of \(\mu\) is 152
2. Calculate the minimum value of \(n\).
   

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)

2. An engineer decided to investigate whether or not the strength of rope was affected by water. A random sample of 9 pieces of rope was taken and each piece was cut in half. One half of each piece was soaked in water over night, and then each piece of rope was tested to find its strength. The results, in coded units, are given in the table below Assuming that the strength of rope follows a normal distribution, test whether or not there is any difference between the mean strengths of dry and wet rope. State your hypotheses clearly and use a \(1 \%\) level of significance.
(8 marks)

5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children. The marks obtained by the two groups are summarised in the table below.

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
   1. the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
   2. the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
      [0pt] [In each case you should give full details of the calculation of the test statistics.]
2. State any assumptions you made in order to carry out these tests.
3. Find a 95\% confidence interval for the common variance of the marks of the two groups.

1 The birth weights, in kilograms, of a random sample of 9 captive-bred elephants are as follows. $$\begin{array} { l l l l l l l l l } 94 & 138 & 130 & 118 & 146 & 165 & 82 & 115 & 69 \end{array}$$ A researcher uses software to produce a \(99 \%\) confidence interval for the mean birth weight of captive-bred elephants. The output from the software is shown in Fig. 1. \begin{table}[h] \end{table}

1. State an assumption about the distribution of the population from which these weights come that is necessary in order to produce this interval.
2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
3. Explain

5 The flight time between two airports is known to be Normally distributed with mean 3.75 hours and standard deviation 0.21 hours. A new airline starts flying the same route. The flight times for a random sample of 12 flights with the new airline are shown in the spreadsheet (Fig. 5), together with the sample mean. \begin{table}[h] \end{table} \section*{(i) In this question you must show detailed reasoning.} You should assume that:

• the flight times for the new airline are Normally distributed,
• the standard deviation of the flight times is still 0.21 hours.

Carry out a test at the \(5 \%\) significance level to investigate whether the mean flight time for the new airline is less than 3.75 hours.
(ii) If both of the assumptions in part (i) were false, name an alternative test that you could carry out to investigate average flight times, stating any assumption necessary for this test.
(iii) If instead the flight times were still Normally distributed but the standard deviation was not known to be 0.21 hours, name another test that you could carry out.

6 A company has a large fleet of cars. It is claimed that use of a fuel additive will reduce fuel consumption. In order to test this claim a researcher at the company randomly selects 40 of the cars. The fuel consumption of each of the cars is measured, both with and without the fuel additive. The researcher then calculates the difference \(d\) litres per kilometre between the two figures for each car, where \(d\) is the fuel consumption without the additive minus the fuel consumption with the additive. The sample mean of \(d\) is 0.29 and the sample standard deviation is 1.64 .

1. Showing your working, find a 95\% confidence interval for the population mean difference.
2. Explain whether the confidence interval suggests that, on average, the fuel additive does reduce fuel consumption.
3. Explain why you can construct the interval in part (i) despite not having any information about the distribution of the population of differences.
4. Explain why the sample used was random.

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 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.

3 A local council collects domestic kitchen waste for composting. Householders place their kitchen waste in a 'compost bin' and this is emptied weekly by the council. The average weight of kitchen waste collected per household each week is known to be 3.4 kg . The council runs a campaign to try to increase the amount of kitchen waste per household which is put in the compost bin. After the campaign, a random sample of 40 households is selected and the weights in kg of kitchen waste in their compost bins are measured. A hypothesis test is carried out in order to investigate whether the campaign has been successful, using software to analyse the sample. The output from the software is shown below.
□
Z Test of a Mean
Null Hypothesis \(\mu = 3.4\) Alternative Hypothesis \(\bigcirc < 0 > 0 \neq\) Sample Result

1. Explain why the test is based on the Normal distribution even though the distribution of the population of amounts of kitchen waste per household is not known.
2. Using the output from the software, complete the test at the \(5 \%\) significance level.
3. Show how the value of \(Z\) in the software output was calculated.
4. Calculate the least value of the sample mean which would have resulted in the conclusion of the test in part (b) being different. You should assume that the standard error is unchanged.

3 The weights in kg of male otters in a large river system are known to be Normally distributed with mean 8.3 and standard deviation 1.8. A researcher believes that weights of male otters in another river are higher because of what he suspects is better availability of food. The researcher records the weights of a random sample of 9 male otters in this other river. The sum of these 9 weights is 83.79 kg .

1. In this question you must show detailed reasoning. You should assume that:
   Show that a test at the \(5 \%\) significance level provides sufficient evidence to conclude that the mean weight of male otters in the other river is greater than 8.3 kg .
2. Explain whether the result of the test suggests that the weights are higher due to better availability of food.
3. If the standard deviation of the weights of otters in the other river could not be assumed to be 1.8 kg , name an alternative test that the researcher could carry out to investigate otter weights.
4. Explain why, even if a test at the \(5 \%\) significance level results in the rejection of the null hypothesis, you cannot be sure that the alternative hypothesis is true.

4 A scientist is investigating sea salinity (the level of salt in the sea) in a particular area. She wishes to check whether satellite measurements, \(y\), of salinity are similar to those directly measured, \(x\). Both variables are measured in parts per thousand in suitable units. The scientist obtains a random sample of 10 values of \(x\) and the related values of \(y\). Below is a screenshot of a scatter diagram to illustrate the data. She decides to carry out a hypothesis test to check if there is any correlation between direct measurement, \(x\), and satellite measurement, \(y\). \includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-5_830_837_589_246}

1. Explain why the scientist might decide to carry out a test based on the product moment correlation coefficient. Summary statistics for \(x\) and \(y\) are as follows. \(n = 10 \quad \sum x = 351.9 \quad \sum y = 350.0 \quad \sum x ^ { 2 } = 12384.5 \quad \sum y ^ { 2 } = 12251.2 \quad \sum \mathrm { xy } = 12317.2\)
2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient.
3. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive correlation between directly measured and satellite measured salinity levels.
4. Explain why it would be preferable to use a larger sample. The scientist is also interested in whether there is any correlation between salinity and numbers of a particular species of shrimp in the water. She takes a large sample and finds that the product moment correlation coefficient for this sample is 0.165 . The result of a test based on this sample is to reject the null hypothesis and conclude that there is correlation between salinity and numbers of shrimp.
5. Comment on the outcome of the hypothesis test with reference to the effect size of 0.165 .

7 A swimming coach believes that times recorded by people using stopwatches are on average 0.2 seconds faster than those recorded by an electronic timing system. In order to test this, the coach takes a random sample of 40 competitors' times recorded by both methods, and finds the differences between the times recorded by the two methods. The mean difference in the times (electronic time minus stopwatch time) is 0.1442 s and the standard deviation of the differences is 0.2580 s .

1. Find a 95\% confidence interval for the mean difference between electronic and stopwatch times.
2. Explain whether there is evidence to suggest that the coach's belief is correct.
3. Explain how you can calculate the confidence interval in part (a) even though you do not know the distribution of the parent population of differences.
4. If the coach wanted to produce a \(95 \%\) confidence interval of width no more than 0.12 s , what is the minimum sample size that would be needed, assuming that the standard deviation remains the same?

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

4 A machine manufactures batches of 100 titanium sheets. The thickness of every sheet in a batch is Normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.03 mm . You should assume that each sheet is of uniform thickness and that the thicknesses of different sheets are independent of each other. The values of \(\mu\) for three different batches, A, B and C, are 3.125, 3.117 and 3.109 respectively.

1. Determine the probability that the total thickness of 10 sheets from Batch A is less than 31.0 mm .
2. Determine the probability that, if a single sheet from Batch A is cut into pieces and 10 of the pieces are stacked together, the total thickness of the stack is less than 31.0 mm .
3. Determine the probability that, if one sheet from each of Batches A, B and C are stacked together, the total thickness of the stack is at least 9.4 mm .
4. Determine the probability that the total thickness of 10 sheets from Batch A is less than the total thickness of 10 sheets from Batch B.