5.05c Hypothesis test: normal distribution for population mean

1. An experiment is conducted to compare the heat retention of two brands of flasks, brand \(A\) and brand \(B\). Both brands of flask have a capacity of 750 ml .

In the experiment 750 ml of boiling water is poured into the flask, which is then sealed. Four hours later the temperature, in \({ } ^ { \circ } \mathrm { C }\), of the water in the flask is recorded. A random sample of 100 flasks from brand \(A\) gives the following summary statistics, where \(x\) is the temperature of the water in the flask after four hours. $$\sum x = 7690 \quad \sum ( x - \bar { x } ) ^ { 2 } = 669.24$$

1. Find unbiased estimates for the mean and variance of the temperature of the water, after four hours, for brand \(A\). A random sample of 80 flasks from brand \(B\) gives the following results, where \(y\) is the temperature of the water in the flask after four hours. $$\bar { y } = 75.9 \quad s _ { y } = 2.2$$
2. Test, at the \(1 \%\) significance level, whether there is a difference in the mean water temperature after four hours between brand \(A\) and brand \(B\). You should state your hypotheses, test statistic and critical value clearly.
3. Explain why it is reasonable to assume that \(\sigma ^ { 2 } = s ^ { 2 }\) in this situation.

1. Roxane, a scientist, carries out an investigation into the fat content of different brands of crisps.

Roxane took random samples of different brands of crisps and recorded, in grams, the fat content ( \(x\) ) of a 30 gram serving. The table below shows some results for just two of these brands.

1. Calculate the value of \(\alpha\) and the value of \(\beta\) Roxane claims that these results show that the crisps from brand A have a lower fat content than the crisps from brand B , as the mean fat content for brand A is, statistically, significantly less than the mean fat content for brand B .
2. Stating your hypotheses clearly, carry out a suitable test, at the \(5 \%\) level of significance, to assess Roxane's claim.
   You should state your test statistic and critical value.
3. For the test in part (b), state whether or not it is necessary to assume that the fat content of crisps is normally distributed. Give a reason for your answer.
4. State an assumption you have made in carrying out the test in part (b).

1. A manager of a large company is investigating the time it takes the company's employees to complete a task.

The manager believes that the mean time for full-time employees to complete the task is more than a minute quicker than the mean time for part-time employees to complete the task. The manager collects a random sample of 605 full-time employees and 45 part-time employees and records the times, \(t\) minutes, it takes each employee to complete the task. The results are summarised in the table below.

1. Test, at the \(5 \%\) level of significance, the manager's claim. You should state your hypotheses, test statistic, critical value and conclusion clearly.
2. State two assumptions you have made in carrying out the test in part (a) The company increases the size of the sample of part-time employees to 46 The time taken to complete the task by the extra employee is 8 minutes.
3. Find an unbiased estimate of the variance for the sample of 46 part-time employees.

5. A greengrocer is investigating the weights of two types of orange, type \(A\) and type \(B\). She believes that on average type \(A\) oranges weigh greater than 5 grams more than type \(B\) oranges. She collects a random sample of 40 type \(A\) oranges and 32 type \(B\) oranges and records the weight, \(x\) grams, of each orange. The table shows a summary of her data.

1. Calculate unbiased estimates for the variance of the weights of the population of type \(A\) oranges and the variance of the weights of the population of type \(B\) oranges.
2. Test, at the \(5 \%\) level of significance, the greengrocer's belief. You should state the hypotheses and the critical value used for this test.
3. Explain how you have used the fact that the sample sizes are large in your answer to part (b).

1. Assam produces bags of flour. The stated weight printed on the bags of flour is 3 kg . The weights of the bags of flour are normally distributed with standard deviation 0.015 kg .

Assam weighs a random sample of 9 bags of flour and finds their mean weight is 2.977 kg .

1. Calculate the \(99 \%\) confidence interval for the mean weight of a bag of flour. Give your limits to 3 decimal places. Assam decides to increase the amount of flour put into the bags.
2. Explain why the confidence interval has led Assam to take this action. After the increase a random sample of \(n\) bags of flour is taken. The sample mean weight of these \(n\) bags is 2.995 kg . A \(95 \%\) confidence interval for \(\mu\) gave a lower limit of less than 2.991 kg .
3. Find the maximum value of \(n\).
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6. Amala believes that the resting heart rate is lower in men who exercise regularly compared to men who do not exercise regularly. She measures the resting heart rate, \(h\), of a random sample of 50 men who exercise regularly and a random sample of 40 men who do not exercise regularly. Her results are summarised in the table below.

1. Calculate the value of \(\alpha\) and the value of \(\beta\)
2. Test, at the \(5 \%\) level of significance, whether there is evidence to support Amala's belief. State your hypotheses clearly.
3. Explain the significance of the central limit theorem to the test in part (b).
4. State two assumptions you have made in carrying out the test in part (b).

6. As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by $$\sum x = 1570 \quad \text { and } \quad \sum x ^ { 2 } = 49467.58$$

1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar { y } = 30.9 \quad \text { and } \quad s _ { y } ^ { 2 } = 3.03$$
2. Test, at the \(5 \%\) level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly.
3. Explain the relevance of the Central Limit Theorem to the test in part (b).
4. State an assumption you have made in carrying out the test in part (b).

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.

A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff.

1. Describe how a stratified sample of 200 staff could be taken.
2. Explain an advantage of using a stratified sample rather than a simple random sample. A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below.

   	Mean score \(( \bar { x } )\)	Variance of scores \(\left( s ^ { 2 } \right)\)
   Full-time staff	52	21
   Part-time staff	50	19

3. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different.
4. Explain the significance of the Central Limit Theorem to the test in part (c).
5. State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic \(z\) was 2.53
6. Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a \(1 \%\) level of significance.
7. Interpret your answers to part (c) and part (f).

5. Upon entering a school, a random sample of eight girls and an independent random sample of eighty boys were given the same examination in mathematics. The girls and boys were then taught in separate classes. After one year, they were all given another common examination in mathematics. The means and standard deviations of the boys' and the girls' marks are shown in the table. You may assume that the test results are normally distributed.

1. Test, at the \(5 \%\) level of significance, whether or not the difference between the means of the boys' and girls' results was significant when they entered school.
2. Test, at the \(5 \%\) level of significance, whether or not the mean mark of the boys is significantly less than the mean mark of the girls in the 'After 1 year' examination.
3. Interpret the results found in part (a) and part (b).

5. A scientist monitored the levels of river pollution near a factory. Before the factory was closed down she took 100 random samples of water from different parts of the river and found an average weight of pollutants of \(10 \mathrm { mg } \mathrm { l } ^ { - 1 }\) with a standard deviation of \(2.64 \mathrm { mg } \mathrm { l } ^ { - 1 }\). After the factory was closed down the scientist collected a further 120 random samples and found that they contained \(8 \mathrm { mg } \mathrm { l } ^ { - 1 }\) of pollutants on average with a standard deviation of \(1.94 \mathrm { mg } \mathrm { l } ^ { - 1 }\). Test, at the \(5 \%\) level of significance, whether or not the mean river pollution fell after the factory closed down.

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.

1. In a trial of \(\operatorname { diet } A\) a random sample of 80 participants were asked to record their weight loss, \(x \mathrm {~kg}\), after their first week of using the diet. The results are summarised by

$$\sum x = 361.6 \text { and } \sum x ^ { 2 } = 1753.95$$

1. Find unbiased estimates for the mean and variance of weight lost after the first week of using diet \(A\). The designers of diet \(A\) believe it can achieve a greater mean weight loss after the first week than a standard diet \(B\). A random sample of 60 people used diet \(B\). After the first week they had achieved a mean weight loss of 4.06 kg , with an unbiased estimate of variance of weight loss of \(2.50 \mathrm {~kg} ^ { 2 }\).
2. Test, at the \(5 \%\) level of significance, whether or not the mean weight loss after the first week using \(\operatorname { diet } A\) is greater than that using diet \(B\). State your hypotheses clearly.
3. Explain the significance of the central limit theorem to the test in part (b).
4. State an assumption you have made in carrying out the test in part (b).
   

1. A sociologist is studying how much junk food teenagers eat. A random sample of 100 female teenagers and an independent random sample of 200 male teenagers were asked to estimate what their weekly expenditure on junk food was. The results are summarised below.

1. Using a 5\% significance level, test whether or not there is a difference in the mean amounts spent on junk food by male teenagers and female teenagers. State your hypotheses clearly.
2. Explain briefly the importance of the central limit theorem in this problem.
   

A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full time staff and 4000 part time staff.

1. Describe how a stratified sample of 200 staff could be taken.
2. Explain an advantage of using a stratified sample rather than a simple random sample. A random sample of 80 full time staff and an independent random sample of 80 part time staff were given a test of policy awareness. The results are summarised in the table below.

   	Mean score \(( \bar { x } )\)	Variance of scores \(\left( s ^ { 2 } \right)\)
   Full time staff	52	21
   Part time staff	50	19

3. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean policy awareness scores for full time and part time staff are different.
4. Explain the significance of the Central Limit Theorem to the test in part (c).
5. State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full time staff and the 80 part time staff were given another test of policy awareness. The value of the test statistic \(z\) was 2.53
6. Comment on the awareness of company policy for the full time and part time staff in light of this result. Use a \(1 \%\) level of significance.
7. Interpret your answers to part (c) and part (f).

5. Mr Alan and Ms Burns are two Mathematics teachers teaching mixed ability groups of students in a large college. At the end of the college year all students took the same examination. A random sample of 29 of Mr Alan's students and a random sample of 26 of Ms Burns' students are chosen. The results are summarised in the table below.

1. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance whether there is evidence that there is a difference in the mean scores of their students. Ms Burns thinks the comparison was unfair as the examination was set by Mr Alan. She looks up a different set of examination results for these students and, although Mr Alan's sample has a higher mean, she calculates the test statistic for this new set of results to be 1.6 However, Mr Alan now claims that the mean marks of his students are higher than the mean marks of Ms Burns' students.
2. Test Mr Alan's claim, stating the hypotheses and critical values you would use. Use a \(10 \%\) level of significance.

7. A farmer monitored the amount of lead in soil in a field next to a factory. He took 100 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(67 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(25 \mathrm { mg } / \mathrm { kg }\).
After the factory closed, the farmer took 150 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(60 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(10 \mathrm { mg } / \mathrm { kg }\).

1. Test at the \(5 \%\) level of significance whether or not the mean weight of lead in the soil decreased after the factory closed. State your hypotheses clearly.
2. Explain the significance of the Central Limit Theorem to the test in part(a).
3. State an assumption you have made to carry out this test.

8. A farmer supplies both duck eggs and chicken eggs. The weights of duck eggs, \(D\) grams, and chicken eggs, \(C\) grams, are such that $$D \sim \mathrm {~N} \left( 54,1.2 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 44,0.8 ^ { 2 } \right)$$

1. Find the probability that the weights of 2 randomly selected duck eggs will differ by more than 3 g .
2. Find the probability that the weight of a randomly selected chicken egg is less than \(\frac { 4 } { 5 }\) of the weight of a randomly selected duck egg. Eggs are packed in boxes which contain either 6 randomly selected duck eggs or 6 randomly selected chicken eggs. The weight of an empty box has distribution \(\mathrm { N } \left( 28 , \sqrt { 5 } ^ { 2 } \right)\).
3. Find the probability that a full box of duck eggs weighs at least 50 g more than a full box of chicken eggs.

6. Fruit-n-Veg4U Market Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .

1. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
2. Explain the relevance of the Central Limit Theorem to the test in part (a).

5. A student believes that there is a difference in the mean lengths of English and French films. He goes to the university video library and randomly selects a sample of 120 English films and a sample of 70 French films. He notes the length, \(x\) minutes, of each of the films in his samples. His data are summarised in the table below.

1. Verify that the unbiased estimate of the variance, \(s ^ { 2 }\), of the lengths of English films is 98.5 minutes \({ } ^ { 2 }\)
2. Stating your hypotheses clearly, test, at the 1\% level of significance, whether or not the mean lengths of English and French films are different.
3. Explain the significance of the Central Limit Theorem to the test in part (b).
4. The university video library contained 724 English films and 473 French films. Explain how the student could have taken a stratified sample of 190 of these films.

2. A researcher believes that the mean weight loss of those people using a slimming plan as part of a group is more than 1.5 kg a year greater than the mean weight loss of those using the plan on their own. The mean weight loss of a random sample of 80 people using the plan as part of a group is 8.7 kg with a standard deviation of 2.1 kg . The mean weight loss of a random sample of 65 people using the plan on their own is 6.6 kg with a standard deviation of 1.4 kg .

1. Stating your hypotheses clearly, test the researcher's claim. Use a \(1 \%\) level of significance.
2. For the test in part (a), state whether or not it is necessary to assume that the weight loss of a person using this plan has a normal distribution. Give a reason for your answer.

3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14 .

1. Describe how this stratified sample should be taken. The weights, \(x \mathrm {~kg}\), of each of the 14 people selected are summarised as $$\sum x = 437 \text { and } \sum x ^ { 2 } = 26983$$
2. Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery's records.
3. Estimate the standard error of the mean. The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.
4. Comment on these values with reference to your answer to part (c) and give a reason for any differences.

6. An engineer has developed a new battery. She claims that the new battery will last more than 8 hours longer, on average, than the old battery. To test the claim, the engineer randomly selects a sample of 50 new batteries and 40 old batteries. She records how long each battery lasts, \(x\) hours for the new batteries and \(y\) hours for the old batteries. The results are summarised in the table below.

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the engineer's claim. State your hypotheses and show your working clearly.
2. Explain the relevance of the Central Limit Theorem to the test in part (a).

Merchandise is sold at concerts. The manager of a concert claims that the mean value of merchandise sold to premium ticket holders is more than \(\pounds 6\) greater than the mean value of merchandise sold to standard ticket holders.

1. Given that all the tickets for the next concert have been sold, describe how a stratified sample should be taken at the concert. The mean value of merchandise sold to a random sample of 60 standard ticket holders at the concert is \(\pounds 15\) with a standard deviation of \(\pounds 10\). The mean value of merchandise sold to a random sample of 55 premium ticket holders at the concert is \(\pounds 23\) with a standard deviation of \(\pounds 8\).
2. Test the manager's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
3. For the test in part (b), state whether or not it is necessary to assume that values of merchandise sold have normal distributions. Give a reason for your answer.
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