5.05c Hypothesis test: normal distribution for population mean

5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows. \(\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}\)

1. It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
   1. Construct a \(98 \%\) confidence interval for the mean weight, \(\mu\) grams, of flour in a bag, giving the limits to four significant figures.
   2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
   3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for \(\mu\) to be valid.
2. The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
3. Given that \(\mu = 1000\), state the probability that a \(98 \%\) confidence interval for \(\mu\) will not contain 1000.
   (l mark)

6 On arrival at a business centre, all visitors are required to register at the reception desk. An analysis of the register, for a random sample of 100 days, results in the following information on the number, \(X\), of visitors per day.

1. Calculate an estimate of:
   1. \(\mu\), the mean number of visitors per day;
   2. \(\sigma\), the standard deviation of the number of visitors per day.
2. Give a reason, based upon the data provided, why \(X\) is unlikely to be normally distributed.
3. 1. Give a reason why \(\bar { X }\), the mean of a random sample of 100 observations on \(X\), may be assumed to be normally distributed.
   2. State, in terms of \(\mu\) and \(\sigma\), the mean and variance of \(\bar { X }\).
4. Hence construct a \(99 \%\) confidence interval for \(\mu\).
5. The receptionist claims that she registers on average more than 30 visitors per day, and frequently registers more than 50 visitors on any one day. Comment on each of these two claims.

4 The weights of packets of sultanas may be assumed to be normally distributed with a standard deviation of 6 grams. The weights of a random sample of 10 packets were as follows: \(\begin{array} { l l l l l l l l l l } 498 & 496 & 499 & 511 & 503 & 505 & 510 & 509 & 513 & 508 \end{array}\)

1. 1. Construct a \(99 \%\) confidence interval for the mean weight of packets of sultanas, giving the limits to one decimal place.
   2. State why, in calculating your confidence interval, use of the Central Limit Theorem was not necessary.
   3. On each packet it states 'Contents 500 grams'. Comment on this statement using both the given sample and your confidence interval.
2. Given that the mean weight of all packets of sultanas is 500 grams, state the probability that a 99\% confidence interval for the mean, calculated from a random sample of packets, will not contain 500 grams.

7

1. A greengrocer displays apples in trays. Each customer selects the apples he or she wishes to buy and puts them into a bag. Records show that the weight of such bags of apples may be modelled by a normal distribution with mean 1.16 kg and standard deviation 0.43 kg . Determine the probability that the mean weight of a random sample of 10 such bags of apples exceeds 1.25 kg .
2. The greengrocer also displays pears in trays. Each customer selects the pears he or she wishes to buy and puts them into a bag. A random sample of 40 such bags of pears had a mean weight of 0.86 kg and a standard deviation of 0.65 kg .
   1. Construct a \(\mathbf { 9 6 \% }\) confidence interval for the mean weight of a bag of pears.
   2. Hence comment on a claim that customers wish to buy, on average, a greater weight of apples than of pears.
      [0pt] [2 marks]

7

1. The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
   Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
2. The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, \(x\) grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
   1. Construct a \(99 \%\) confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
   2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
      Claim 2: All pouches contain more than 300 grams of dry cat food.
      [0pt] [4 marks]
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-24_2288_1705_221_155}

2 A group of estate agents in a particular area claimed that, after the introduction of a new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area had not changed from 8 weeks.

1. A random sample of 9 house purchases in the area revealed that their completion times, in weeks, were as follows. $$\begin{array} { l l l l l l l l l } 6 & 7 & 10 & 12 & 9 & 11 & 7 & 8 & 14 \end{array}$$ Assuming that completion times in the area are normally distributed with standard deviation 2.5 weeks, test, at the \(5 \%\) level of significance, the group's claim. (7 marks)
2. It was subsequently discovered that, after the introduction of the new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area remained at 8 weeks. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out your hypothesis test in part (a). Give a reason for your answer.
   (2 marks)

3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 . After having the series of lessons with David, Becki decides to investigate his claim.
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, \(x\), that she takes to complete the round. Her results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$ Using a \(z\)-test and the \(1 \%\) level of significance, investigate David's claim.

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.

1 A machine fills bottles with bleach. The volume, in millilitres, of bleach dispensed by the machine into a bottle may be modelled by a normal distribution with mean \(\mu\) and standard deviation 8 . A recent inspection indicated that the value of \(\mu\) was 768 . Yvonne, the machine's operator, claims that this value has not subsequently changed. Zara, the quality control supervisor, records the volume of bleach in each of a random sample of 18 bottles filled by the machine and calculates their mean to be 764.8 ml . Test, at the \(5 \%\) level of significance, Yvonne's claim that the mean volume of bleach dispensed by the machine has not changed from 768 ml .

6 Bishen believes that the mean weight of boxes of black peppercorns is 45 grams. Abi, thinking that this is not the case, weighs, in grams, a random sample of 8 boxes of black peppercorns, with the following results. $$\begin{array} { l l l l l l l l } 44 & 44 & 43 & 46 & 42 & 40 & 43 & 46 \end{array}$$

1. 1. Construct a \(95 \%\) confidence interval for the mean weight of boxes of black peppercorns, stating any assumption that you make.
   2. Comment on Bishen's belief.
2. 1. Abi claims that the mean weight of boxes of black peppercorns is less than 45 grams. Test this claim at the \(5 \%\) level of significance.
   2. If Bishen's belief is true, state, with a reason, what type of error, if any, may have occurred when conclusions to the test in part (b)(i) were drawn.
      (2 marks)

6 Alex obtained the actual waist measurements, \(w\) inches, of a random sample of 50 pairs of jeans, each of which was labelled as having a 32 -inch waist. The results are summarised by $$n = 50 , \quad \Sigma w = 1615.0 , \quad \Sigma w ^ { 2 } = 52214.50$$ Test, at the \(0.1 \%\) significance level, whether this sample provides evidence that the mean waist measurement of jeans labelled as having 32 -inch waists is in fact greater than 32 inches. State your hypotheses clearly. \section*{Jan 2006}

7 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). The mean of a random sample of 12 observations of \(X\) is denoted by \(\bar { X }\). A test is carried out at the \(1 \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 80\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu < 80\). The test is summarised as follows: 'Reject \(\mathrm { H } _ { 0 }\) if \(\bar { X } < c\); otherwise do not reject \(\mathrm { H } _ { 0 } { } ^ { \prime }\).

1. Calculate the value of \(c\).
2. Assuming that \(\mu = 80\), state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
   1. \(\bar { X } = 74.0\),
   2. \(\bar { X } = 75.0\).
   3. Independent repetitions of the above test, using the value of \(c\) found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of \(\mu\).

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

1. Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
2. Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
   1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a \(98 \%\) confidence interval for the average number of letters per week received at work by Rosa.
   2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

7 A shop sells cooked chickens in two sizes: medium and large.
The weights, \(X\) grams, of medium chickens may be assumed to be normally distributed with mean \(\mu _ { X }\) and standard deviation 45. The weights, \(Y\) grams, of large chickens may be assumed to be normally distributed with mean \(\mu _ { Y }\) and standard deviation 65. A random sample of 20 medium chickens had a mean weight, \(\bar { x }\) grams, of 936 .
A random sample of 10 large chickens had the following weights in grams: $$\begin{array} { l l l l l l l l l l } 1165 & 1202 & 1077 & 1144 & 1195 & 1275 & 1136 & 1215 & 1233 & 1288 \end{array}$$

1. Calculate the mean weight, \(\bar { y }\) grams, of this sample of large chickens.
2. Hence investigate, at the \(1 \%\) level of significance, the claim that the mean weight of large chickens exceeds that of medium chickens by more than 200 grams.
3. 1. Deduce that, for your test in part (b), the critical value of \(( \bar { y } - \bar { x } )\) is 253.24, correct to two decimal places.
   2. Hence determine the power of your test in part (b), given that \(\mu _ { Y } - \mu _ { X } = 275\).
   3. Interpret, in the context of this question, the value that you obtained in part (c)(ii).
      (3 marks)

3 Kutz and Styler are two unisex hair salons. An analysis of a random sample of 150 customers at Kutz shows that 28 per cent are male. An analysis of an independent random sample of 250 customers at Styler shows that 34 per cent are male.

1. Test, at the \(5 \%\) level of significance, the hypothesis that there is no difference between the proportion of male customers at Kutz and that at Styler.
2. State, with a reason, the probability of making a Type I error in the test in part (a) if, in fact, the actual difference between the two proportions is 0.05 .

7 In a town, the total number, \(R\), of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.

1. Using the \(10 \%\) level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of \(R\).
2. Determine, for your test in part (a), the critical region for \(R\).
3. Assuming that the offer for sale of houses on the new housing development has reduced the mean value of \(R\) to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
   (4 marks)

4 The waiting times for patients to see a doctor in a hospital can be modelled with a normal distribution with known variance of 10 minutes. 4

1. A random sample of 100 patients has a total waiting time of 3540 minutes.
   Calculate a \(98 \%\) confidence interval for the population mean of waiting times, giving values to four significant figures.
   4
2. Dante conducts a hypothesis test with the sample from part (a) on the waiting times. Dante's hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 38 \\ & \mathrm { H } _ { 1 } : \mu \neq 38 \end{aligned}$$ Dante uses a \(2 \%\) level of significance.
   Explain whether Dante accepts or rejects the null hypothesis.

4 The height of lilac trees, in metres, can be modelled by a normal distribution with variance 0.7 A random sample of \(n\) lilac trees is taken and used to construct a 99\% confidence interval for the population mean. This confidence interval is \(( 5.239,5.429 )\) 4

1. Find the value of \(n\) 4
2. Joey claims that the mean height of lilac trees is 5.3 metres.
   State, with a reason, whether the confidence interval supports Joey's claim.

6 The number of computers sold per day by a shop can be modelled by the random variable \(Y\) where \(Y \sim \operatorname { Po } ( 42 )\) 6

1. State the variance of \(Y\) 6
2. One month ago, the shop started selling a new model of computer.
   On a randomly chosen day in the last month, the shop sold 53 computers.
   Carry out a hypothesis test, at the \(5 \%\) level of significance, to investigate whether the mean number of computers sold per day has increased in the last month.
   [0pt] [6 marks]
   6
3. Describe, in the context of the hypothesis test in part (b), what is meant by a Type II error.

5 Rebekah is investigating the distances, \(X\) light years, between the Earth and visible stars in the night sky. She determines the distance between the Earth and a star for a random sample of 100 visible stars. The summarised results are as follows: $$\sum x = 35522 \quad \text { and } \quad \sum x ^ { 2 } = 32902257$$ 5

1. Calculate a 97\% confidence interval for the population mean of \(X\), giving your values to the nearest light year.
   5
2. Mike claims that the population mean is 267 light years. Rebekah says that the confidence interval supports Mike's claim. State, with a reason, whether Rebekah is correct.

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.

5 The mass, \(X\), in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5

1. A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is \(( 81.7,91.3 )\) with values correct to three significant figures.
   5
2. Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the \(1 \%\) level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85 \\ & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
   5
3. Interpret, in context, the conclusion to the hypothesis test in part (b).

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.