One-sample t-test, variance unknown

7 A farmer grows a particular type of fruit tree. On average, the mass of fruit produced per tree has been 6.2 kg . He has developed a new kind of soil and claims that the mean mass of fruit produced per tree when growing in this new soil has increased. A random sample of 10 trees grown in the new soil is chosen. The masses, \(x \mathrm {~kg}\), of fruit produced are summarised as follows. $$\Sigma x = 72.0 \quad \Sigma x ^ { 2 } = 542.0$$ Test at the \(5 \%\) significance level whether the farmer's claim is justified, assuming a normal distribution.

7 A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows. $$\begin{array} { l l l l l l l } 98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2 \end{array}$$ Assuming that masses are normally distributed, test, at the \(10 \%\) significance level, whether the mean mass of athletes in this competition is equal to 94 kg .

9 A farmer grows large amounts of a certain crop. On average, the yield per plant has been 0.75 kg . The farmer has improved the soil in which the crop grows, and she claims that the yield per plant has increased. A random sample of 10 plants grown in the improved soil is chosen. The yields, \(x \mathrm {~kg}\), are summarised as follows. $$\Sigma x = 7.85 \quad \Sigma x ^ { 2 } = 6.19$$

1. Test at the \(5 \%\) significance level whether the farmer's claim is justified, assuming a normal distribution.
2. Find a 95\% confidence interval for the population mean yield for plants grown in the new soil.

7 A random sample of 10 observations of a normally distributed random variable \(X\) gave the following summarised data, where \(\bar { x }\) denotes the sample mean. $$\Sigma x = 70.4 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 8.48$$ Test, at the \(10 \%\) significance level, whether the population mean of \(X\) is less than 7.5.

9 A random sample of 9 observations of a normally distributed random variable \(X\) gave the following summarised data. $$\Sigma x = 94.5 \quad \Sigma x ^ { 2 } = 993.6$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is 10.2 . Calculate a \(90 \%\) confidence interval for the population mean of \(X\).

6 A random sample of 8 observations of a normal random variable \(X\) has mean \(\bar { x }\), where $$\bar { x } = 6.246 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.784$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is less than 6.44.

9 There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows. $$\begin{array} { l l l l l l l l } 1.75 & 1.72 & 1.62 & 1.70 & 1.82 & 1.75 & 1.68 & 1.84 \end{array}$$ You may assume that heights of students are normally distributed.

1. Test, at the \(5 \%\) significance level, whether the population mean height of students at this college is greater than 1.70 metres.
2. Find a \(95 \%\) confidence interval for the population mean height of students at this college.

3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean \(380 ^ { \circ } \mathrm { C }\). The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.

1. Give three reasons why a \(t\) test would be appropriate.
2. Carry out the test using a \(5 \%\) significance level. State your hypotheses and conclusion carefully.
3. Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
4. Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.

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.

15 Bottles of Fizzipop nominally contain 330 ml of drink. A consumer affairs researcher collects a random sample of 55 bottles of Fizzipop and records the volume of drink in each bottle. Summary statistics for the researcher's sample are shown in the table.

1. 1. Calculate the mean volume of drink in a bottle of Fizzipop.
   2. Show that the standard deviation of the volume of drink in a bottle of Fizzipop is 3.78 ml . The researcher uses software to produce a histogram with equal class intervals, which is shown below. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-10_533_759_1181_251}
2. Explain why the researcher decides that the Normal distribution is a suitable model for the volume of drink in a bottle of Fizzipop.
3. Use your answers to parts (a) and (b) to determine the expected number of bottles which contain less than 330 ml in a random sample of 100 bottles. In order to comply with new regulations, no more than 1\% of bottles of Fizzipop should contain less than 330 ml . The manufacturer decides to meet the new regulations by adjusting the manufacturing process so that the mean volume of drink in a bottle of Fizzipop is increased. The standard deviation is unaltered.
4. Determine the minimum mean volume of drink in a bottle of Fizzipop which should ensure that the new regulations are met. Give your answer to \(\mathbf { 3 }\) significant figures. The mean volume of drink in a bottle of Fizzipop is set to 340 ml . After several weeks the quality control manager suspects the mean volume may have reduced. She collects a random sample of 100 bottles of Fizzipop. The mean volume of drink in a bottle in the sample is found to be 339.37 ml .
5. Assuming the standard deviation is unaltered, conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the mean volume of drink in a bottle of Fizzipop is less than 340 ml .

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.

15 A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.

1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes.
2. Use the data in the table to estimate
   • the sample mean,
   • the sample standard deviation.
   The data in the table on the previous page are represented in the following histogram, Fig 15: \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-10_728_1577_312_315} \captionsetup{labelformat=empty} \caption{Fig. 15}
   \end{figure} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).
3. Comment briefly on whether the histogram supports this choice of model.
4. 1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
   2. Compare your answer with your answer to part (a). The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.
5. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the \(5 \%\) level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\mathrm { H } _ { 0 } : \mu = 210 , \mathrm { H } _ { 1 } : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4.
   [0pt] [5]

6 The random variable \(X\) was assumed to have a normal distribution with mean \(\mu\). Using a random sample of size 128, a significance test was carried out using the following hypotheses. \(\mathrm { H } _ { 0 } : \mu = 30\) \(\mathrm { H } _ { 1 } : \mu > 30\) It was found that \(\sum x = 3929.6\) and \(\sum x ^ { 2 } = 123483.52\). The conclusion of the test was to reject the null hypothesis.

1. Determine the range of possible values of the significance level of the test.
2. It was subsequently found that \(X\) was not normally distributed. Explain whether this invalidates the conclusion of the test.

8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the \(5 \%\) level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.

5 Jasmine's French teacher states that a homework assignment should take, on average, 30 minutes to complete. Jasmine believes that he is understating the mean time that the assignment takes to complete and so decides to investigate. She records the times, in minutes, that it takes for a random sample of 10 students to complete the French assignment, with the following results: $$\begin{array} { l l l l l l l l l l } 29 & 33 & 36 & 42 & 30 & 28 & 31 & 34 & 37 & 35 \end{array}$$

1. Test, at the \(1 \%\) level of significance, Jasmine's belief that her French teacher has understated the mean time that it should take to complete the homework assignment.
2. State an assumption that you must make in order for the test used in part (a) to be valid.

3 Lorraine bought a new golf club. She then practised with this club by using it to hit golf balls on a golf range. After several such practice sessions, she believed that there had been no change from 190 metres in the mean distance that she had achieved when using her old club. To investigate this belief, she measured, at her next practice session, the distance, \(x\) metres, of each of a random sample of 10 shots with her new club. Her results gave $$\sum x = 1840 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1240$$ Investigate Lorraine's belief at the \(2 \%\) level of significance, stating any assumption that you make.
(7 marks)

6 The contents, in millilitres, of cartons of milk produced at Kream Dairies, can be modelled by a normal distribution with mean 568 and variance \(\sigma ^ { 2 }\). After receiving several complaints from their customers who thought that the average content of the cartons had been reduced, the production manager of Kream Dairies decided to investigate. A random sample of 8 cartons of milk was taken, revealing the following contents, in millilitres. $$\begin{array} { l l l l l l l l } 560 & 568 & 561 & 562 & 564 & 567 & 565 & 563 \end{array}$$ Investigate, at the \(1 \%\) level of significance, whether the average content of cartons of milk is less than 568 millilitres.
(10 marks)

6 The lifetime, \(X\) hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.

1. The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows: $$\begin{array} { l l l l l } 85 & 92 & 100 & 95 & 99 \end{array}$$ Test the members' suspicion at the \(5 \%\) level of significance.
2. The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, \(x\) hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where \(\bar { x }\) denotes the sample mean: $$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$ Test the manufacturer's belief at the \(5 \%\) level of significance.

6 The management of the Wellfit gym claims that the mean cholesterol level of those members who have held membership of the gym for more than one year is 3.8 . A local doctor believes that the management's claim is too low and investigates by measuring the cholesterol levels of a random sample of 7 such members of the Wellfit gym, with the following results: $$\begin{array} { l l l l l l l } 4.2 & 4.3 & 3.9 & 3.8 & 3.6 & 4.8 & 4.1 \end{array}$$ Is there evidence, at the \(5 \%\) level of significance, to justify the doctor's belief that the mean cholesterol level is greater than the management's claim? State any assumption that you make.

1 Judith, the village postmistress, believes that, since moving the post office counter into the local pharmacy, the mean daily number of customers that she serves has increased from 79. In order to investigate her belief, she counts the number of customers that she serves on 12 randomly selected days, with the following results. $$\begin{array} { l l l l l l l l l l l l } 88 & 81 & 84 & 89 & 90 & 77 & 72 & 80 & 82 & 81 & 75 & 85 \end{array}$$ Stating a necessary distributional assumption, test Judith's belief at the 5\% level of significance. \begin{verbatim} QUESTION PART REFERENCE \end{verbatim}

6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.

1. William weighs a random sample of 6 pears, obtaining the following weights, in grams. $$\begin{array} { l l l l l l } 160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8 \end{array}$$ Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the \(5 \%\) level of significance supports William's suspicion.
   (7 marks)
2. William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and \(s = 11.2\) grams, where \(s ^ { 2 }\) is an unbiased estimate of the population variance. Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the \(\mathbf { 1 \% }\) level of significance.
3. Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).

6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.

1. Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the \(10 \%\) significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
   [0pt] [8 marks]
2. Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the \(5 \%\) significance level. Give your answer to two decimal places.
   [0pt] [4 marks] \(7 \quad\) A continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
3. Find \(\mathrm { P } ( X < 1 )\).
4. 1. Show that, for \(1 \leqslant x \leqslant 3\), the cumulative distribution function, \(\mathrm { F } ( x )\), is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
   2. Hence verify that the median value of \(X\) lies between 1.13 and 1.14 .
      [0pt] [3 marks] QUESTION
      PART Answer space for question 7
      REFERENCE REFERENCE
      \includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-20_2290_1707_221_153}

2. A mechanic is required to change car tyres. An inspector timed a random sample of 20 tyre changes and calculated the unbiased estimate of the population variance to be 6.25 minutes \({ } ^ { 2 }\). Test, at the \(5 \%\) significance level, whether or not the standard deviation of the population of times taken by the mechanic is greater than 2 minutes. State your hypotheses clearly.
(6)

3. A machine is set to fill bags with flour such that the mean weight is 1010 grams. To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows. $$\begin{array} { l l l l l l l l } 1010 & 1015 & 1005 & 1000 & 998 & 1008 & 1012 & 1007 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.)
(Total 8 marks)

1. Historical records from a large colony of squirrels show that the weight of squirrels is normally distributed with a mean of 1012 g . Following a change in the diet of squirrels, a biologist is interested in whether or not the mean weight has changed.

A random sample of 14 squirrels is weighed and their weights \(x\), in grams, recorded. The results are summarised as follows: $$\Sigma x = 13700 , \quad \Sigma x ^ { 2 } = 13448750 .$$ Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a change in the mean weight of the squirrels.