5.05c Hypothesis test: normal distribution for population mean

3 The random variable \(X\) is the length of time in minutes that Jannon takes to mend a bicycle puncture. \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is given that \(\mathrm { P } ( X > 30.0 ) = 0.1480\) and \(\mathrm { P } ( X > 20.9 ) = 0.6228\). Find \(\mu\) and \(\sigma\).

6 The lengths of body feathers of a particular species of bird are modelled by a normal distribution. A researcher measures the lengths of a random sample of 600 body feathers from birds of this species and finds that 63 are less than 6 cm long and 155 are more than 12 cm long.

1. Find estimates of the mean and standard deviation of the lengths of body feathers of birds of this species.
2. In a random sample of 1000 body feathers from birds of this species, how many would the researcher expect to find with lengths more than 1 standard deviation from the mean?

6 Last year, the mean time taken by students at a school to complete a certain test was 25 minutes. Akash believes that the mean time taken by this year's students was less than 25 minutes. In order to test this belief, he takes a large random sample of this year's students and he notes the time taken by each student. He carries out a test, at the \(2.5 \%\) significance level, for the population mean time, \(\mu\) minutes. Akash uses the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 25\).

1. Give a reason why Akash should use a one-tailed test.
   Akash finds that the value of the test statistic is \(z = - 2.02\).
2. Explain what conclusion he should draw.
   In a different one-tailed hypothesis test the \(z\)-value was found to be 2.14 .
3. Given that this value would lead to a rejection of the null hypothesis at the \(\alpha \%\) significance level, find the set of possible values of \(\alpha\).
   The population mean time taken by students at another school to complete a test last year was \(m\) minutes. Sorin carries out a one-tailed test to determine whether the population mean this year is less than \(m\), using a random sample of 100 students. He assumes that the population standard deviation of the times is 3.9 minutes. The sample mean is 24.8 minutes, and this result just leads to the rejection of the null hypothesis at the 5\% significance level.
4. Find the value of \(m\).
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2 In the past the yield of a certain crop, in tonnes per hectare, had mean 0.56 and standard deviation 0.08 Following the introduction of a new fertilizer, the farmer intends to test at the \(2.5 \%\) significance level whether the mean yield has increased. He finds that the mean yield over 10 years is 0.61 tonnes per hectare.

1. State two assumptions that are necessary for the test.
2. Carry out the test.

4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.

1. Show that \(\operatorname { Var } ( X ) = 1.2\).
   The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found.
2. Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
3. Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
4. Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 . Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.

4 The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave \(\Sigma x = 2130\) and \(\Sigma x ^ { 2 } = 37746\), where \(x\) is the time in minutes to mark an examination paper.

1. Calculate unbiased estimates of the population mean and variance.
2. Stating the null and alternative hypotheses, use a \(10 \%\) significance level to test whether the examination board's estimated time is consistent with the data.

3 A consumer group, interested in the mean fat content of a particular type of sausage, takes a random sample of 20 sausages and sends them away to be analysed. The percentage of fat in each sausage is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 26 & 27 & 28 & 28 & 28 & 29 & 29 & 30 & 30 & 31 & 32 & 32 & 32 & 33 & 33 & 34 & 34 & 34 & 35 & 35 \end{array}$$ Assume that the percentage of fat is normally distributed with mean \(\mu\), and that the standard deviation is known to be 3 .

1. Calculate a 98\% confidence interval for the population mean percentage of fat.
2. The manufacturer claims that the mean percentage of fat in sausages of this type is 30 . Use your answer to part (i) to determine whether the consumer group should accept this claim.

7 Machine \(A\) fills bags of fertiliser so that their weights follow a normal distribution with mean 20.05 kg and standard deviation 0.15 kg . Machine \(B\) fills bags of fertiliser so that their weights follow a normal distribution with mean 20.05 kg and standard deviation 0.27 kg .

1. Find the probability that the total weight of a random sample of 20 bags filled by machine \(A\) is at least 2 kg more than the total weight of a random sample of 20 bags filled by machine \(B\). [6]
2. A random sample of \(n\) bags filled by machine \(A\) is taken. The probability that the sample mean weight of the bags is greater than 20.07 kg is denoted by \(p\). Find the value of \(n\), given that \(p = 0.0250\) correct to 4 decimal places.

3 The number of customers who visit a particular shop between 9.00 am and 10.00 am has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 5.2. Following some new advertising, the manager wishes to test whether the value of \(\lambda\) has increased. He chooses a random sample of 20 days and finds that the total number of customers who visited the shop between 9.00 am and 10.00 am on those days is 125 . Use an approximating distribution to test at the \(2.5 \%\) significance level whether the value of \(\lambda\) has increased.

7 A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on a particular day.

1. He considers choosing the first 50 customers who visit the information desk. Explain why this method is unsuitable.
   The actual lengths of time, in minutes, that customers spend at the information desk may be assumed to have mean \(\mu\) and variance 4.8. The researcher knows that in the past the value of \(\mu\) was 6.0. He wishes to test, at the \(2 \%\) significance level, whether this is still true. He chooses a random sample of 50 customers and notes how long they each spend at the information desk.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.
4. Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).
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5 The time, in minutes, spent by customers at a particular gym has the distribution \(\mathrm { N } ( \mu , 38.2 )\). In the past the value of \(\mu\) has been 42.4. Following the installation of some new equipment the management wishes to test whether the value of \(\mu\) has changed.

1. State what is meant by a Type I error in this context.
2. The mean time for a sample of 20 customers is found to be 45.6 minutes. Test at the \(2.5 \%\) significance level whether the value of \(\mu\) has changed.

2 In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation 0.12 . Following the introduction of some new signals, it is required to test whether the mean journey time has decreased.

1. State what is meant by a Type II error in this context.
2. The mean time for a random sample of 50 journeys is found to be 1.36 hours. Assuming that the standard deviation of journey times is still 0.12 hours, test at the \(2.5 \%\) significance level whether the population mean journey time has decreased.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (b).

3 The local council claims that the average number of accidents per year on a particular road is 0.8 . Jane claims that the true average is greater than 0.8 . She looks at the records for a random sample of 3 recent years and finds that the total number of accidents during those 3 years was 5 .

1. Assume that the number of accidents per year follows a Poisson distribution.
   1. State null and alternative hypotheses for a test of Jane's claim.
   2. Test at the \(5 \%\) significance level whether Jane's claim is justified.
2. Jane finds that the number of accidents per year has been gradually increasing over recent years. State how this might affect the validity of the test carried out in part (a)(ii).

7 In the past, the mean time for Jenny's morning run was 28.2 minutes. She does some extra training and she wishes to test whether her mean time has been reduced. After the training Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is less than 27 minutes she will conclude that the training has been effective. You may assume that, after the training, Jenny's run time has a standard deviation of 4.0 minutes.

1. State suitable null and alternative hypotheses for Jenny's test.
2. Find the probability that Jenny will make a Type I error.
3. Jenny found that the sample mean run time was 27.2 minutes. Explain briefly whether it is possible for her to make a Type I error or a Type II error or both.
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2 In the past, the mean height of plants of a particular species has been 2.3 m . A random sample of 60 plants of this species was treated with fertiliser and the mean height of these 60 plants was found to be 2.4 m . Assume that the standard deviation of the heights of plants treated with fertiliser is 0.4 m . Carry out a test at the \(2.5 \%\) significance level of whether the mean height of plants treated with fertiliser is greater than 2.3 m .

3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.

1. Give a reason for using a sample rather than the whole population in carrying out this test.
   A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.

3 In the past, the annual amount of wheat produced per farm by a large number of similar sized farms in a certain region had mean 24.0 tonnes and standard deviation 5.2 tonnes. Last summer a new fertiliser was used by all the farms, and it was expected that the mean amount of wheat produced per farm would be greater than 24.0 tonnes. In order to test whether this was true, a scientist recorded the amounts of wheat produced by a random sample of 50 farms last summer. He found that the value of the sample mean was 25.8 tonnes. Stating a necessary assumption, carry out the test at the \(1 \%\) significance level.

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.

5 Last year the mean time for pizza deliveries from Pete's Pizza Pit was 32.4 minutes. This year the time, \(t\) minutes, for pizza deliveries from Pete's Pizza Pit was recorded for a random sample of 50 deliveries. The results were as follows. $$n = 50 \quad \Sigma t = 1700 \quad \Sigma t ^ { 2 } = 59050$$

1. Find unbiased estimates of the population mean and variance.
2. Test, at the \(2 \%\) significance level, whether the mean delivery time has changed since last year.
3. Under what circumstances would it not be necessary to use the Central Limit Theorem in answering (b)?

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
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6 The masses of cereal boxes filled by a certain machine have mean 510 grams. An adjustment is made to the machine and an inspector wishes to test whether the mean mass of cereal boxes filled by the machine has decreased. After the adjustment is made, he chooses a random sample of 120 cereal boxes. The mean mass of these boxes is found to be 508 grams. Assume that the standard deviation of the masses is 10 grams.

1. Test at the \(2.5 \%\) significance level whether the mean mass of cereal boxes filled by the machine has decreased.
   Later the inspector carries out a similar test at the \(2.5 \%\) significance level, using the same hypotheses and another 120 randomly chosen cereal boxes.
   [0pt]
2. Given that the mean mass is now actually 506 grams, find the probability of a Type II error. [5]

6 The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.

1. Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.
   The manufacturers of Frutos claim that the mean number of green sweets in a packet is 1.65 .
   Anji believes that the true value of the mean, \(\mu\), is less than 1.65 . She uses the results from the 200 randomly chosen packets to test the manufacturers' claim.
2. State suitable null and alternative hypotheses for the test. \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-08_2714_37_143_2008}
3. Show that the result of Anji's test is significant at the \(5 \%\) level but not at the \(1 \%\) level.
4. It is given that Anji made a Type I error. Explain how this shows that the significance level that Anji used in her test was not \(1 \%\).

3 In the past, the mean time taken by Freda for a particular daily journey was 39.2 minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, \(t\) minutes, for 40 randomly chosen journeys and summarises the results as follows. $$n = 40 \quad \Sigma t = 1504 \quad \Sigma t ^ { 2 } = 57760$$

1. Calculate unbiased estimates of the population mean and variance of the new journey time.
2. Test, at the \(5 \%\) significance level, whether the population mean time has decreased.

3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, \(z\), and finds that \(z = 2.41\).

1. Explain briefly whether he should use a one-tail test or a two-tail test.
2. Carry out the test at the \(1 \%\) significance level.

6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
   Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
3. Find the probability of a Type I error.
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