Two-sample t-test with summary statistics

A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).

6 A random sample of 50 observations of a random variable \(X\) and a random sample of 60 observations of a random variable \(Y\) are taken. The results for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates for the population variances, \(s _ { x } ^ { 2 }\) and \(s _ { y } ^ { 2 }\), respectively, are as follows. $$\bar { x } = 25.4 \quad \bar { y } = 23.6 \quad s _ { x } ^ { 2 } = 23.2 \quad s _ { y } ^ { 2 } = 27.8$$ A test, at the \(\alpha \%\) significance level, of the null hypothesis that the population means of \(X\) and \(Y\) are equal against the alternative hypothesis that they are not equal is carried out. Given that the null hypothesis is not rejected, find the set of possible values of \(\alpha\).

Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$\begin{array} { l l l l l l l l } 15.3 & 12.0 & 15.1 & 11.2 & 14.4 & 13.8 & 12.4 & 11.8 \end{array}$$ Assuming a normal distribution, find a \(95 \%\) confidence interval for the mean length of fish in \(A\). The lengths of the 6 fish from \(B\), in the same units, are as follows. $$\begin{array} { l l l l l l } 15.0 & 10.7 & 13.6 & 12.4 & 11.6 & 12.6 \end{array}$$ Stating any assumptions that you make, test at the \(5 \%\) significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). Calculate a 95\% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\).

A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x \mathrm {~kg}\). The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y \mathrm {~kg}\). The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$ Test, at the \(5 \%\) significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer B's Crown plants. Test, at the \(5 \%\) significance level, whether Farmer A's claim is justified.

8 The weekly salaries of employees at two large electronics companies, \(A\) and \(B\), are being compared. The weekly salaries of an employee from company \(A\) and an employee from company \(B\) are denoted by \(\\) x\( and \)\\( y\) respectively. A random sample of 50 employees from company \(A\) and a random sample of 40 employees from company \(B\) give the following summarised data. $$\Sigma x = 5120 \quad \Sigma x ^ { 2 } = 531000 \quad \Sigma y = 3760 \quad \Sigma y ^ { 2 } = 375135$$

1. The population mean salaries of employees from companies \(A\) and \(B\) are denoted by \(\\) \mu _ { A }\( and \)\\( \mu _ { B }\) respectively. Using a \(5 \%\) significance level, test the null hypothesis \(\mu _ { A } = \mu _ { B }\) against the alternative hypothesis \(\mu _ { A } \neq \mu _ { B }\).
2. State, with a reason, whether any assumptions about the distributions of employees' salaries are needed for the test in part (i).

In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. $$\Sigma x = 10.56 \quad \Sigma x ^ { 2 } = 14.1775 \quad \Sigma y = 12.39 \quad \Sigma y ^ { 2 } = 15.894$$ A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\).

1. Test, at the \(10 \%\) significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal.
   A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as $$\Sigma x = 10.24 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.294$$ where \(\bar { x }\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p \mathrm {~kg}\). A test of this claim is carried out at the \(10 \%\) significance level, using only this second sample from lake \(A\). This test supports the scientist's claim.
2. Find the greatest possible value of \(p\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 A random sample of 8 elephants from region \(A\) is taken and their weights, \(x\) tonnes, are recorded. ( 1 tonne \(= 1000 \mathrm {~kg}\).) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x ^ { 2 } = 131.82$$ A random sample of 10 elephants from region \(B\) is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of 0.1555 tonnes \({ } ^ { 2 }\). The distributions of the weights of elephants in regions \(A\) and \(B\) are both assumed to be normal with the same population variance. Test at the \(10 \%\) significance level whether the mean weight of elephants in region \(A\) is the same as the mean weight of elephants in region \(B\).

A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x \mathrm {~kg}\). The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y \mathrm {~kg}\). The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$

1. Test, at the \(5 \%\) significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make.
   A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\) 's Crown plants.
2. Test, at the \(5 \%\) significance level, whether Farmer \(A\) 's claim is justified.

5 Claire grows strawberries on her farm. She wants to compare two brands of fertiliser, brand \(A\) and brand \(B\). She grows two sets of plants of the same variety of strawberries under the same conditions, fertilising one set with brand \(A\) and the other with brand \(B\). The yields per plant, in grams, from each set of plants are summarised below.

1. Stating your hypotheses clearly, carry out a suitable test to assess whether the mean yield from plants using fertiliser \(A\) is greater than the mean yield from plants using fertiliser \(B\).
   Use a 1\% level of significance and state your test statistic and critical value. The total cost of fertiliser \(A\) for Claire's 50 plants was \(\pounds 75\) The total cost of fertiliser \(B\) for Claire's 40 plants was \(\pounds 50\) Claire sells all her strawberries at \(\pounds 3\) per kilogram.
2. Use this information, together with your answer in part (a), to advise Claire on which of the two brands of fertiliser she should use next year in order to maximise her expected profit per plant, giving a reason for your answer.

3. A grocer believes that the average weight of a grapefruit from farm \(A\) is greater than the average weight of a grapefruit from farm \(B\). The weights, in grams, of 80 grapefruit selected at random from farm \(A\) have a mean value of 532 g and a standard deviation, \(s _ { A }\), of 35 g . A random sample of 100 grapefruit from farm \(B\) have a mean weight of 520 g and a standard deviation, \(s _ { B }\), of 28 g . Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the grocer's belief is supported by the data.

1. 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).

5. A dance studio has 800 dancers of which \begin{displayquote} 452 are beginners
251 are intermediates
97 are professionals

1. Explain in detail how a stratified sample of size 50 could be taken.
2. State an advantage of stratified sampling rather than simple random sampling in this situation. \end{displayquote} Independent random samples of 80 beginners and 60 intermediates are chosen. Each of these dancers is given an assessment score, \(x\), based on the quality of their dancing. The results are summarised in the table below.

   	\(\bar { x }\)	\(s ^ { 2 }\)	\(n\)
   Beginners	31.7	57.3	80
   Intermediates	36.9	38.1	60

   The studio manager believes that the mean score of intermediates is more than 3 points greater than the mean score of beginners.
3. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not these data support the studio manager's belief.

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).

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).

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.

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 shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is \(\pounds x\). A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.

1. Find the values of \(\bar { x }\) and \(s ^ { 2 }\).
2. Test, at the \(5 \%\) level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly.

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.

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.

5. A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.

1. Test, at the \(5 \%\) level of significance, the doctor's claim. State your hypotheses clearly.
2. State any assumptions you have made in testing the doctor's claim. The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
3. Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures.

3. As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet.
2. Explain briefly any assumptions you made in order to carry out this test.

4 Holly, a horticultural researcher, believes that the mean height of stems on Tahiti daffodils exceeds that on Jetfire daffodils by more than 15 cm . She measures the heights, \(x\) centimetres, of stems on a random sample of 65 Tahiti daffodils and finds that their mean, \(\bar { x }\), is 40.7 and that their standard deviation, \(s _ { x }\), is 3.4 . She also measures the heights, \(y\) centimetres, of stems on a random sample of 75 Jetfire daffodils and finds that their mean, \(\bar { y }\), is 24.4 and that their standard deviation, \(s _ { y }\), is 2.8 . Investigate, at the \(1 \%\) level of significance, Holly's belief.

2 As part of a comparison of two varieties of cucumber, Fanfare and Marketmore, random samples of harvested cucumbers of each variety were selected and their lengths measured, in centimetres. The results are summarised in the table.

1. Test, at the \(1 \%\) level of significance, the hypothesis that there is no difference between the mean length of harvested Fanfare cucumbers and that of harvested Marketmore cucumbers.
2. In addition to length, name one other characteristic of cucumbers that could be used for comparative purposes.