2.05e Hypothesis test for normal mean: known variance

1. The heights of females from a country are normally distributed with

• a mean of 166.5 cm
• a standard deviation of 6.1 cm

Given that \(1 \%\) of females from this country are shorter than \(k \mathrm {~cm}\),

1. find the value of \(k\)
2. Find the proportion of females from this country with heights between 150 cm and 175 cm A female, from this country, is chosen at random from those with heights between 150 cm and 175 cm
3. Find the probability that her height is more than 160 cm The heights of females from a different country are normally distributed with a standard deviation of 7.4 cm Mia believes that the mean height of females from this country is less than 166.5 cm
   Mia takes a random sample of 50 females from this country and finds the mean of her sample is 164.6 cm
4. Carry out a suitable test to assess Mia's belief. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   \section*{Question 5 continued.} \section*{Question 5 continued.} \section*{Question 5 continued.}

13 Each weekday Keira drives to work with her son Kaito. She always sets off at 8.00 a.m. She models her journey time, \(x\) minutes, by the distribution \(X \sim \mathrm {~N} ( 15,4 )\). Over a long period of time she notes that her journey takes less than 14 minutes on \(7 \%\) of the journeys, and takes more than 18 minutes on \(31 \%\) of the journeys.

1. Investigate whether Keira's model is a good fit for the data. Kaito believes that Keira's value for the variance is correct, but realises that the mean is not correct.
2. Find, correct to two significant figures, the value of the mean that Keira should use in a refined model which does fit the data. Keira buys a new car. After driving to work in it each day for several weeks, she randomly selects the journey times for \(n\) of these days. Her mean journey time for these \(n\) days is 16 minutes. Using the refined model she conducts a hypothesis test to see if her mean journey time has changed, and finds that the result is significant at the \(5 \%\) level.
3. Determine the smallest possible value of \(n\).

10 Club 65-80 Holidays fly jets between Liverpool and Magaluf. Over a long period of time records show that half of the flights from Liverpool to Magaluf take less than 153 minutes and \(5 \%\) of the flights take more than 183 minutes. An operations manager believes that flight times from Liverpool to Magaluf may be modelled by the Normal distribution.

1. Use the information above to write down the mean time the operations manager will use in his Normal model for flight times from Liverpool to Magaluf.
2. Use the information above to find the standard deviation the operations manager will use in his Normal model for flight times from Liverpool to Magaluf, giving your answer correct to 1 decimal place.
3. Data is available for 452 flights. A flight time of under 2 hours was recorded in 16 of these flights. Use your answers to parts (a) and (b) to determine whether the model is consistent with this data. The operations manager suspects that the mean time for the journey from Magaluf to Liverpool is less than from Liverpool to Magaluf. He collects a random sample of 24 flight times from Magaluf to Liverpool. He finds that the mean flight time is 143.6 minutes.
4. Use the Normal model used in part (c) to conduct a hypothesis test to determine whether there is evidence at the \(1 \%\) level to suggest that the mean flight time from Magaluf to Liverpool is less than the mean flight time from Liverpool to Magaluf.
   [0pt]
5. Identify two ways in which the Normal model for flight times from Liverpool to Magaluf might be adapted to provide a better model for the flight times from Magaluf to Liverpool. [2]

13 A large supermarket chain advertises that the mean mass of apples of a certain variety on sale in their stores is 0.14 kg . Following a poor growing season, the head of quality control believes that the mean mass of these apples is less than 0.14 kg and she decides to carry out a hypothesis test at the \(5 \%\) level of significance. She collects a random sample of this variety of apple from the supermarket chain and records the mass, in kg, of each apple. She uses software to analyse the data. The results are summarised in the output below.

1. State the null hypothesis and the alternative hypothesis for the test, defining the parameter used.
2. Write down the distribution of the sample mean for this hypothesis test.
3. Determine the critical region for the test.
4. Carry out the test, giving your conclusion in context.

9 A company supplies computers to businesses. In the past the company has found that computers are kept by businesses for a mean time of 5 years before being replaced. Claud, the manager of the company, thinks that the mean time before replacing computers is now different.

1. Describe how Claud could obtain a cluster sample of 120 computers used by businesses the company supplies. Claud decides to conduct a hypothesis test at the \(5 \%\) level to test whether there is evidence to suggest that the mean time that businesses keep computers is not 5 years. He takes a random sample of 120 computers. Summary statistics for the length of time computers in this sample are kept are shown in Fig. 9. \begin{table}[h]

   Statistics
   \(n\)	120
   Mean	4.8855
   \(\sigma\)	2.6941
   \(s\)	2.7054
   \(\Sigma x\)	586.2566
   \(\Sigma x ^ { 2 }\)	3735.1475
   Min	0.1213
   Q1	2.5472
   Median	4.8692
   Q3	7.0349
   Max	9.9856

   \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{table} \section*{(b) In this question you must show detailed reasoning.}
   • State the hypotheses for this test, explaining why the alternative hypothesis takes the form it does.
   • Use a suitable distribution to carry out the test.

11 In 2010 the heights of adult women in the UK were found to have mean \(\mu = 161.6 \mathrm {~cm}\) and variance \(\sigma ^ { 2 } = 1.96 \mathrm {~cm} ^ { 2 }\). It is believed that the mean height of adult women in 2020 in the UK is greater than in 2010. In 2020 a researcher collected a random sample of the heights of 200 adult women in the UK. The researcher calculated the sample mean height and carried out a hypothesis test at the \(5 \%\) level to investigate whether there was any evidence to suggest that the mean height of adult women in the UK had increased. The researcher assumed that the variance was unaltered.

1. - State suitable hypotheses for the test, defining any variables you use.
   The researcher found that the sample mean was 161.9 cm and made the following statements.

3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8 The shop has 9 Birdscope cameras in stock at the start of each week. A week is selected at random.

1. Find the probability that the demand for Birdscope cameras cannot be met in this particular week. In a year, there are 50 weeks in which Birdscope cameras can be sold.
2. Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
3. Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2 The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold. Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
4. use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.

1. A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .

A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm . Use a suitable test, at the \(5 \%\) level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.

4. A manufacturing company produces solar panels. The output of each solar panel is normally distributed with standard deviation 6 watts. It is thought that the mean output, \(\mu\), is 160 watts. A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the \(5 \%\) level of significance. He tests \(\mathrm { H } _ { 0 } : \mu = 160\) against \(\mathrm { H } _ { 1 } : \mu > 160\) On reporting to his manager, the researcher can only find 4 of the output values. These are shown below $$\begin{array} { l l l l } 168.2 & 157.4 & 173.3 & 161.1 \end{array}$$ Given that the result of the hypothesis test is that there is significant evidence to reject \(\mathrm { H } _ { 0 }\) at the \(5 \%\) level of significance, calculate the minimum possible missing output value, \(\alpha\). Give your answer correct to 1 decimal place.

7. A machine fills packets with \(X\) grams of powder where \(X\) is normally distributed with mean \(\mu\). Each packet is supposed to contain 1 kg of powder. To comply with regulations, the weight of powder in a randomly selected packet should be such that \(\mathrm { P } ( X < \mu - 30 ) = 0.0005\)

1. Show that this requires the standard deviation to be 9.117 g to 3 decimal places. A random sample of 10 packets is selected from the machine. The weight, in grams, of powder in each packet is as follows 999.8991 .61000 .31006 .11008 .2997 .0993 .21000 .0997 .11002 .1
2. Assuming that the standard deviation of the population is 9.117 g , test, at the \(1 \%\) significance level, whether or not the machine is delivering packets with mean weight of less than 1 kg . State your hypotheses clearly.

2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly. The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable \(B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)\). The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable \(G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)\). The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg . Test, at the \(5 \%\) level of significance, the hypothesis that \(\mu _ { B } = \mu _ { G }\).
[0pt] [6 marks]

3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches. Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.

1. Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the \(5 \%\) level of significance. The total height of the females in the sample is 832 ft .
2. Carry out the test making your conclusion clear.

2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, \(x\), and fungi, \(y\), in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between \(x\) and \(y\). \begin{figure}[h] \end{figure}

1. Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data. Summary statistics for the data are as follows. \(n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800\)
2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient between \(x\) and \(y\).
3. Carry out a test at the \(5 \%\) significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
4. Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.

3 A student is investigating the link between temperature (in degrees Celsius) and electricity consumption (in Gigawatt-hours) in the country in which he lives. The student has read that there is strong negative correlation between daily mean temperature over the whole country and daily electricity consumption during a year. He wonders if this applies to an individual season. He therefore obtains data on the mean temperature and electricity consumption on ten randomly selected days in the summer. The spreadsheet output below shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-3_798_1593_639_251}

1. Calculate Pearson's product moment correlation coefficient between daily mean temperature and daily electricity consumption. The student decides to carry out a hypothesis test to investigate whether there is negative correlation between daily mean temperature and daily electricity consumption during the summer.
2. Explain why the student decides to carry out a test based on Pearson's product moment correlation coefficient.
3. Show that the test at the \(5 \%\) significance level does not result in the null hypothesis being rejected.
4. The student concludes that there is no correlation between the variables in the summer months. Comment on the student's conclusion.

10 A researcher is investigating the actual lengths of time that patients spend at their appointments with the doctors at a certain clinic. There are 12 doctors at the clinic, and each doctor has 24 appointments per day. The researcher plans to choose a sample of 24 appointments on a particular day.

1. The researcher considers the following two methods for choosing the sample. Method A: Choose a random sample of 24 appointments from the 288 on that day.
   Method B: Choose one doctor's 1st and 2nd appointments. Choose another doctor's 3rd and 4th appointments and so on until the last doctor's 23rd and 24th appointments. For each of A and B state a disadvantage of using this method. Appointments are scheduled to last 10 minutes. The researcher suspects that the actual times that patients spend are more than 10 minutes on average. To test this suspicion, he uses method A , and takes a random sample of 24 appointments. He notes the actual time spent for each appointment and carries out a hypothesis test at the \(1 \%\) significance level.
2. Explain why a 1-tail test is appropriate. The population mean of the actual times that patients spend at their appointments is denoted by \(\mu\) minutes.
3. Assuming that \(\mu = 10\), state the probability that the conclusion of the test will be that \(\mu\) is not greater than 10 . The actual lengths of time, in minutes, that patients spend for their appointments may be assumed to have a normal distribution with standard deviation 3.4.
   [0pt]
4. Given that the total length of time spent for the 24 appointments is 285 minutes, carry out the test. [7]
5. In part (iv) it was necessary to use the fact that the sample mean is normally distributed. Give a reason why you know that this is true in this case.

12 In the past, the time spent by customers in a certain shop had mean 10.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 12.0 minutes.

1. Assuming that the standard deviation is unchanged, test at the \(1 \%\) significance level whether the mean time spent by customers in the shop has changed.
2. Another random sample of 50 customers is chosen and a similar test at the \(1 \%\) significance level is carried out. Given that the population mean time has not changed, state the probability that the conclusion of the test will be that the population mean time has changed.

11 In the past the masses of new-born babies in a certain country were normally distributed with mean 3300 g . Last year a publicity campaign was held to encourage pregnant women to improve their diet. Following this campaign, it is required to test whether the mean mass of new-born babies has increased. A random sample of 200 new-born babies is chosen, and it is found that their mean mass is 3360 g . It is given that the standard deviation of the masses of new-born babies is 450 g . Carry out the test at the 2.5\% significance level.

1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.

Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.

1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.

1. A machine cuts strips of metal to length \(L \mathrm {~cm}\), where \(L\) is normally distributed with standard deviation 0.5 cm .

Strips with length either less than 49 cm or greater than 50.75 cm cannot be used.
Given that 2.5\% of the cut lengths exceed 50.98 cm ,

1. find the probability that a randomly chosen strip of metal can be used. Ten strips of metal are selected at random.
2. Find the probability fewer than 4 of these strips cannot be used. A second machine cuts strips of metal of length \(X \mathrm {~cm}\), where \(X\) is normally distributed with standard deviation 0.6 cm A random sample of 15 strips cut by this second machine was found to have a mean length of 50.4 cm
3. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test whether or not the mean length of all the strips, cut by the second machine, is greater than 50.1 cm

1. Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.

$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .

1. Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
2. Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
3. Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
4. test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
   (You may assume that the mass of the contents of a packet is normally distributed.)
5. Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
   (Total 14 marks)

8 A company records the number of complaints, \(X\), that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8

1. Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
   8
2. The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
   The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the \(10 \%\) level of significance.
   8
3. It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places. \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-15_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-20_2496_1723_214_148}

7 A newspaper reports that the average price of unleaded petrol in the UK is 110.2 p per litre. The price, in pence, of a litre of unleaded petrol at a random sample of 15 petrol stations in Yorkshire is shown below together with some output from software used to analyse the data. \begin{table}[h] \end{table}

1. Select a suitable hypothesis test to investigate whether there is any evidence that the average price of unleaded petrol in Yorkshire is different from 110.2 p. Justify your choice of test.
2. Conduct the hypothesis test at the \(5 \%\) level of significance.