One-sample z-test, variance known

3 The masses, in kilograms, of newborn babies in country \(A\) are represented by the random variable \(X\), with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The masses of a random sample of 500 newborn babies in this country were found and the results are summarised below. $$n = 500 \quad \Sigma x = 1625 \quad \Sigma x ^ { 2 } = 5663.5$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   A researcher wishes to test whether the mean mass of newborn babies in a neighbouring country, \(B\), is different from that in country \(A\). He chooses a random sample of 60 newborn babies in country \(B\) and finds that their sample mean mass is 2.95 kg . Assume that your unbiased estimates in part (a) are the correct values for \(\mu\) and \(\sigma ^ { 2 }\). Assume also that the variance of the masses of newborn babies in country \(B\) is the same as in country \(A\).
2. Carry out the test at the \(1 \%\) significance level.

4 The mean mass of packets of sugar is supposed to be 505 g . A random sample of 10 packets filled by a certain machine was taken and the masses, in grams, were found to be as follows. $$\begin{array} { l l l l l l l l l l } 500 & 499 & 496 & 495 & 498 & 490 & 492 & 501 & 494 & 494 \end{array}$$

1. Find unbiased estimates of the population mean and variance.
   The mean mass of packets produced by this machine was found to be less than 505 g , so the machine was adjusted. Following the adjustment, the masses of a random sample of 150 packets from the machine were measured and the total mass was found to be 75660 g .
2. Given that the population standard deviation is 3.6 g , test at the \(2 \%\) significance level whether the machine is still producing packets with mean mass less than 505 g .
3. Explain why the use of the normal distribution is justified in carrying out the test in part (ii). [1]

1

1. The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.
2. Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles.
3. Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic.
4. Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is \(10 \%\) of the value of sales in that month. The value, in \(\pounds\), of the monthly sales has the distribution \(\mathrm { N } \left( 21200,1100 ^ { 2 } \right)\). Find the probability that a randomly chosen claim lies between \(\pounds 3000\) and \(\pounds 3300\). William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results. $$\begin{array} { l l l l l l l l l } 1.046 & 1.048 & 1.039 & 1.055 & 1.038 & 1.054 & 1.038 & 1.051 & 1.038 \end{array}$$
5. William has to test whether the specific gravity of the mixture meets the requirement. Why might a \(t\) test be used for these data and what assumption must be made?
6. Carry out the test using a significance level of \(10 \%\).
7. Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a \(95 \%\) confidence interval.

9 A random sample of five metal rods produced by a machine is taken. Each rod is tested for hardness. The results, in suitable units, are as follows. $$\begin{array} { l l l l l } 524 & 526 & 520 & 523 & 530 \end{array}$$ Assuming a normal distribution, calculate a \(95 \%\) confidence interval for the population mean. Some adjustments are made to the machine. Assume that a normal distribution is still appropriate and that the population variance remains unchanged. A second random sample, this time of ten metal rods, is now taken. The results for hardness are as follows. $$\begin{array} { l l l l l l l l l l } 525 & 520 & 522 & 524 & 518 & 520 & 519 & 525 & 527 & 516 \end{array}$$ Stating suitable hypotheses, test at the \(10 \%\) significance level whether there is any difference between the population means before and after the adjustments.

10 The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 . This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the 5\% significance level, whether the mean level of pollutant has changed.

10 In the past, the time spent in minutes, by customers in a certain library had mean 32.5 and standard deviation 8.2. Following a change of layout in the library, the mean time spent in the library by a random sample of 50 customers is found to be 34.5 minutes. Assuming that the standard deviation remains at 8.2 , test at the \(5 \%\) significance level whether the mean time spent by customers in the library has changed.

14 A survey during 2013 investigated mean expenditure on bread and on alcohol.
The 2013 survey obtained information from 12144 adults.
The survey revealed that the mean expenditure per adult per week on bread was 127p.
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1. For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p. 14
2. 1. Carry out a hypothesis test, at the \(5 \%\) significance level, to investigate whether the mean expenditure per adult per week on bread changed from 2012 to 2013. Assume that the survey data is a random sample taken from a normal distribution.
      [0pt] [5 marks] 14
3. (ii) Calculate the greatest and least values for the sample mean expenditure on bread per adult per week for 2013 that would have resulted in acceptance of the null hypothesis for the test you carried out in part (a)(i). Give your answers to two decimal places.
   [0pt] [2 marks] 14
4. The 2013 survey revealed that the mean expenditure per adult, per week on alcohol was 324p. The mean expenditure per adult per week on alcohol for 2009 was 307p.
   A test was carried out on the following hypotheses relating to mean expenditure per adult per week on alcohol in 2013. \(\mathrm { H } _ { 0 } : \mu = 307\) \(\mathrm { H } _ { 1 } : \mu \neq 307\) This test resulted in the null hypothesis, \(\mathrm { H } _ { 0 }\), being rejected.
   State, with a reason, whether the test result supports the following statements:
   14
5. 1. the mean UK expenditure on alcohol per adult per week increased by 17 p from 2009 to 2013; 14
6. (ii) the mean UK consumption of alcohol per adult per week changed from 2009 to 2013.

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 \(n\).

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.
   • Explain whether the researcher conducted a 1-tail or a 2-tail test.
   • Determine the critical region for the test.
   The researcher found that the sample mean was 161.9 cm and made the following statements.
   • The sample mean is in the critical region.
   • The null hypothesis is accepted.
   • This proves that the mean height of adult women in the UK is unaltered at 161.6 cm .
   • Explain whether each of these statements is correct.

4 The greatest weight \(W N\) that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80 . A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N .

1. Test at the \(1 \%\) significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support.
2. State an assumption needed in carrying out the test in part (a).
3. Explain whether it is necessary to use the central limit theorem in carrying out the test.

7 Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$\begin{aligned} n & = 49 \\ \Sigma x & = 74.48 \\ \Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$ Test, at the \(5 \%\) significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .

3. The manager of a gym claimed that the mean age of its customers is 30 years. A random sample of 75 customers is taken and their ages have a mean of 28.2 years and a standard deviation, \(s\), of 8.5 years.

1. Stating your hypotheses clearly and using a 10\% level of significance, test whether or not the manager's claim is supported by the data.
2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
3. State an additional assumption needed to carry out the test in part (a).

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.

1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.

Test, at the \(5 \%\) level of significance, the managing director's claim. State your hypotheses clearly.

3. It is known from past evidence that the weight of coffee dispensed into jars by machine \(A\) is normally distributed with mean \(\mu _ { \mathrm { A } }\) and standard deviation 2.5 g . Machine \(B\) is known to dispense the same nominal weight of coffee into jars with mean \(\mu _ { B }\) and standard deviation 2.3 g . A random sample of 10 jars filled by machine \(A\) contained a mean weight of 249 g of coffee. A random sample of 15 jars filled by machine \(B\) contained a mean weight of 251 g .

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the population mean weight dispensed by machine B is greater than that of machine A .
2. Write down an assumption needed to carry out this test.

1. The time, in minutes, it takes Robert to complete the puzzle in his morning newspaper each day is normally distributed with mean 18 and standard deviation 3. After taking a holiday, Robert records the times taken to complete a random sample of 15 puzzles and he finds that the mean time is 16.5 minutes. You may assume that the holiday has not changed the standard deviation of times taken to complete the puzzle.

Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a reduction in the mean time Robert takes to complete the puzzle.

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.

6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 . Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving. In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.

1. Investigate, at the \(5 \%\) level of significance, the teachers' suspicion.
2. Explain, in the context of this question, the meaning of a Type I error.

3 The handicap committee of a golf club has indicated that the mean score achieved by the club's members in the past was 85.9 . A group of members believes that recent changes to the golf course have led to a change in the mean score achieved by the club's members and decides to investigate this belief. A random sample of the scores, \(x\), of 100 club members was taken and is summarised by $$\sum x = 8350 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 15321$$ where \(\bar { x }\) denotes the sample mean.
Test, at the \(5 \%\) level of significance, the group's belief that the mean score of 85.9 has changed.

1 David claims that customers have to queue at a supermarket checkout for more than 5 minutes, on average. The queuing times, \(x\) minutes, of 40 randomly selected customers result in \(\bar { x } = 5.5\) and \(s ^ { 2 } = 1.31\). Investigate, at the \(1 \%\) level of significance, David's claim.

1 Roger claims that, on average, his journey time from home to work each day is greater than 45 minutes. The times, \(x\) minutes, of 30 randomly selected journeys result in \(\bar { x } = 45.8\) and \(s ^ { 2 } = 4.8\).
Investigate Roger's claim at the \(1 \%\) level of significance.

5 In 2001, the mean height of students at the end of their final year at Bright Hope Secondary School was 165 centimetres. In 2010, David and James selected a random sample of 100 students who were at the end of their final year at this school. They recorded these students' heights, \(x\) centimetres, and found that \(\bar { x } = 167.1\) and \(s ^ { 2 } = 101.2\). To investigate the claim that the mean height had increased since 2001, David and James each correctly conducted a hypothesis test. They used the same null hypothesis and the same alternative hypothesis. However, David used a \(5 \%\) level of significance whilst James used a \(1 \%\) level of significance.

1. 1. Write down the null and alternative hypotheses that both David and James used.
      (l mark)
   2. Determine the outcome of each of the two hypothesis tests, giving each conclusion in context.
   3. State why both David and James made use of the Central Limit Theorem in their hypothesis tests.
2. It was later found that, in 2010, the mean height of students at the end of their final year at Bright Hope Secondary School was actually 165 centimetres. Giving a reason for your answer in each case, determine whether a Type I error or a Type II error or neither was made in the hypothesis test conducted by:
   1. David;
   2. James.