Hypothesis testing: exam review Exam review: second paper

A tablet maker states that the mean time to charge a tablet fully is 90 minutes. A tester believes that the mean time is greater than this. He will test his belief at the 5% significance level. [5]

1. (a) State the null hypothesis and the alternative hypothesis. Define the symbol you use. [2]
2. (b) He tests a random sample of 20 tablets and finds a sample mean of 95 minutes. If the maker's statement is true, the probability of a sample mean of 95 minutes or more is 0.19. State whether the tester has sufficient evidence to support his belief, and give a reason. [3]

A plant breeder predicts that the seeds from a cross will be round yellow, round green, wrinkled yellow and wrinkled green in the ratio 9 : 3 : 3 : 1. She grows 320 seeds and records the type of each. The results are in the table. A goodness of fit test is used to test her prediction. [8]

1. (a) State the null hypothesis and the alternative hypothesis. [1]
2. (b) Find the expected frequency for each type of seed. [2]
3. (c) Write down the number of degrees of freedom. [1]
4. (d) Use your calculator to find the \(\chi^2\) statistic and the \(p\)-value. [2]
5. (e) Test at the 5% significance level and state your conclusion in context. [2]

A spinner has eight equal sectors and is spun 160 times. A goodness of fit test, at the 1% significance level, of the null hypothesis that the spinner is fair gives \(\chi^2=10.3\). [6]

1. (a) Write down the number of degrees of freedom. [1]
2. (b) Find the critical value of the test at the 1% significance level. [2]
3. (c) Compare the test statistic with the critical value and state your conclusion in context. [2]
4. (d) Find the \(p\)-value of the test. [1]

A call centre states that the mean time to answer a call is 40 seconds. A manager believes that the mean time is less than this since new staff were hired. She records the time, in seconds, to answer each of 12 randomly chosen calls: 39.3, 40.5, 37.9, 40.2, 41.8, 40.4, 43.6, 36.7, 39.4, 37.2, 37.0, 38.7. [8]

1. (a) State the null hypothesis and the alternative hypothesis. Define the symbol you use. [2]
2. (b) State an assumption that is needed for the test to be valid. [1]
3. (c) Use your calculator to find the mean and the standard deviation \(s_{n-1}\) of the sample. [1]
4. (d) Use your calculator to find the \(t\)-statistic and the \(p\)-value for the test. [2]
5. (e) Conclude at the 5% significance level, in context. [2]

A teacher compares two revision methods. 10 students use method 1 and have a mean test score of 82.4 and \(s_{n-1}=6.2\). 14 different students use method 2 and have a mean test score of 79.1 and \(s_{n-1}=5.8\). A test is carried out to see whether the mean scores of students using the two methods are different. [9]

1. (a) State two assumptions that are needed for the pooled two-sample \(t\)-test. [2]
2. (b) State the null hypothesis and the alternative hypothesis. Define the symbols you use. [2]
3. (c) Write down the number of degrees of freedom. [1]
4. (d) Use your calculator to find the \(t\)-statistic and the \(p\)-value for the test. [2]
5. (e) Conclude at the 5% significance level, in context. [2]

Two tasters each rank eight cheeses, A to H, from best to worst. Taster 1 orders the cheeses B, G, D, A, E, H, C, F. Taster 2 orders them D, A, H, B, C, E, G, F. A one-tailed test is used, at the 5% significance level, to see whether the tasters tend to agree. The critical value is 0.643. [6]

1. (a) Write down the rank that each taster gave to each cheese, for cheeses A to H. [2]
2. (b) Use your calculator to find Spearman's rank correlation coefficient, \(r_s\). [1]
3. (c) State the null hypothesis and the alternative hypothesis. [1]
4. (d) Conclude at the 5% significance level, in context. [2]

A game uses a fair six-sided die. In each of 400 turns a player rolls the die 4 times and records \(X\), the number of sixes. The results are in the table. A goodness of fit test is used to test whether \(X\sim B(4, \frac{1}{6})\). [13]

1. (a) State the null hypothesis and the alternative hypothesis. [2]
2. (b) Find the expected frequencies for \(X=0\), \(X=1\), \(X=2\), \(X=3\) and \(X=4\). [3]
3. (c) Explain why \(X=4\) cannot be kept as a separate group. [1]
4. (d) The groups are combined so that every expected frequency is at least 5. Write down the number of degrees of freedom. [1]
5. (e) Use your calculator to find the \(\chi^2\) statistic and the \(p\)-value. [2]
6. (f) Test at the 5% significance level and state your conclusion in context. [2]
7. (g) Find which group contributes most to the \(\chi^2\) statistic, and say whether it has more or fewer observations than expected. [2]

A school asks 154 students how they travel to school and whether they arrived late on a given day. The results are in the table. A \(\chi^2\) test for independence is used to test whether arriving late is independent of the way a student travels. [13]

1. (a) State the null hypothesis and the alternative hypothesis. [2]
2. (b) Write down the number of degrees of freedom. [1]
3. (c) Show how to find the expected frequency for Walking and Late. [2]
4. (d) Use your calculator to find the \(\chi^2\) statistic and the \(p\)-value. [2]
5. (e) Test at the 5% significance level and state your conclusion in context. [2]
6. (f) Find the cell with the greatest contribution to the \(\chi^2\) statistic, and compare its observed and expected frequencies. [2]
7. (g) State the conclusion if the significance level is 1%. [2]

Nine flats are at different distances from a city centre. The distance from the centre, \(x\) km, and the monthly rent, \(y\) euros, of each flat are recorded. The values of \(x\) are 1.2, 2.5, 3.1, 4.8, 6.0, 6.0, 9.5, 11.0, 18.2 and the values of \(y\) are 640, 720, 900, 640, 800, 560, 700, 780, 610. A test is carried out, at the 5% significance level, to see whether rent tends to be lower further from the centre. The critical value for this one-tailed test is 0.6. [14]

1. (a) Write down the ranks of the values of \(x\) and of the values of \(y\), giving tied values their average rank. [3]
2. (b) Use your calculator to find Spearman's rank correlation coefficient, \(r_s\). [1]
3. (c) State the null hypothesis and the alternative hypothesis. [2]
4. (d) Use the critical value to conclude, at the 5% significance level, in context. [3]
5. (e) Interpret the value of \(r_s\) in the context of the question. [2]
6. (f) The largest value of \(x\), 18.2, was recorded wrongly. The correct value is 16.0. Without calculating, state how this changes \(r_s\), and give a reason. [2]
7. (g) The values of \(y\) are converted from euros to cents. State how this changes \(r_s\). [1]

A gym asks 209 members which time of day they usually train and which type of membership they hold. A \(\chi^2\) test for independence is to be used to test whether the time of day is independent of membership type. [10]

1. (a) Show how to find the expected frequency for Morning and Corporate. [2]
2. (b) Write down one expected frequency that is less than 5, and explain why Family and Corporate should be combined. Write down the table of observed frequencies after the columns are combined. [3]
3. (c) Write down the number of degrees of freedom for the test on the combined table. [1]
4. (d) Use your calculator to find the \(\chi^2\) statistic and the \(p\)-value for the combined table, and state the conclusion at the 5% significance level, in context. [4]

A website records the number of server errors in each of 100 hours. The results are in the table. The owner thinks that the number of errors in an hour follows a Poisson distribution, and uses a \(\chi^2\) goodness of fit test with the mean estimated from the data. [10]

1. (a) State the null hypothesis, and find the mean number of errors in an hour. [2]
2. (b) Using this mean, show that the expected frequency of 2 errors is 24.2, correct to 3 significant figures. [2]
3. (c) The expected frequency of exactly 4 errors is 3.95. Explain why the groups 4, 5 and 6 are combined with any larger value into one group, and state the number of degrees of freedom. [2]
4. (d) Use your calculator to find the \(\chi^2\) statistic and the \(p\)-value, and state the conclusion at the 5% significance level, in context. [4]

A flour mill states that the mean mass of a sack of flour is 25.0 kg. A trading standards officer believes that the mean mass is greater than this. The mass of a sack is normally distributed with standard deviation 0.40 kg, which is known. A random sample of 50 sacks has a mean mass of 25.11 kg. [7]

1. (a) Write down the null and alternative hypotheses, and state, with a reason, whether a \(z\)-test or a \(t\)-test should be used. [2]
2. (b) Find the \(p\)-value for the test. [2]
3. (c) The officer then tests a second mill, where the standard deviation is not known. The masses, in kg, of a random sample of 10 sacks from this mill are: 25.2, 24.9, 25.3, 25.1, 25.4, 24.8, 25.2, 25.3, 25.0, 25.35. Assuming that these masses are normally distributed, test the same belief for this mill. Find the \(p\)-value and state the conclusion at the 5% significance level, in context. [3]

A courier compares two routes, 1 and 2, between the same two depots. She times 45 journeys on route 1 and 40 different journeys on route 2. The journey times are normally distributed. The population standard deviations are known to be 4.2 minutes for route 1 and 3.9 minutes for route 2. The mean time on route 1 is 36.4 minutes and the mean time on route 2 is 35.2 minutes. She believes that route 1 takes longer. [8]

1. (a) State the null hypothesis and the alternative hypothesis, defining the symbols you use. [2]
2. (b) Use your calculator to find the \(p\)-value for a two-sample \(z\)-test. [2]
3. (c) Conclude at the 5% significance level, in context. [2]
4. (d) Suppose that the population standard deviations were not known, but were assumed to be equal. State the test that would be used, and two assumptions that are needed for it. [2]

Nine students sit a practice test and, after a revision course, a second test of the same difficulty. Their scores are in the table. A teacher believes that the revision course increases the mean score. [8]

1. (a) State the null hypothesis and the alternative hypothesis, taking \(d\) as the second value minus the first value and defining \(\mu_d\). [2]
2. (b) State one advantage of using a paired test rather than a two-sample test here. [1]
3. (c) Use your calculator to find the \(t\)-statistic and the \(p\)-value for the test. [2]
4. (d) Conclude at the 5% significance level, in context. [2]
5. (e) State an assumption that is needed for the test to be valid. [1]

A farm compares two feeds, A and B, for chickens. A random sample of 50 chickens given feed A has a mean mass gain of 620 g over a month. An independent random sample of 60 chickens given feed B has a mean mass gain of 566 g. The population standard deviations are known to be 70 g for feed A and 64 g for feed B. The distributions of the mass gains are not known. The supplier claims that feed A gives a mean mass gain more than 30 g greater than feed B. Let \(\mu_A\) and \(\mu_B\) be the population mean mass gains for feeds A and B. [8]

1. (a) Write down the distribution of \(\bar X_A-\bar X_B\) in terms of \(\mu_A\) and \(\mu_B\). [2]
2. (b) Explain why this distribution can be used even though the distributions of the individual times are not known. [1]
3. (c) Write down the null and alternative hypotheses. [1]
4. (d) Find the \(p\)-value for the test. [2]
5. (e) State the conclusion at the 5% significance level, in context. [2]

A machine cuts lengths of cable. The length of a cable, \(X\) cm, is normally distributed with standard deviation 0.8 cm, and the length is claimed to have mean 150 cm. An inspector takes a random sample of 25 cables and tests, at the 5% significance level, whether the mean length differs from 150 cm. Let \(\mu\) be the population mean length. [9]

1. (a) Write down the distribution of \(\bar X\), the mean of the sample of 25, when \(H_0\) is true. [1]
2. (b) Find the critical region for \(\bar X\) at the 5% significance level. [3]
3. (c) The sample mean is 150.18 cm. State the conclusion of the test, in context. [1]
4. (d) A second inspector rejects \(H_0\) if \(\bar X\) differs from 150 by more than 0.25 cm. Find the probability that this inspector makes a Type I error. [2]
5. (e) The true value of \(\mu\) is in fact 150.4. Using the critical region in part (b), find the probability of a Type II error. [2]

A cafe manager says that 15% of customers order a decaffeinated drink. After a health campaign the owner believes that the proportion has increased. She asks 40 customers, chosen at random, and counts the number \(X\) who order a decaffeinated drink. She tests her belief at the 5% significance level. [13]

1. (a) Write down suitable null and alternative hypotheses, defining the parameter you use. [2]
2. (b) Find the critical region for the test at the 5% significance level. [3]
3. (c) Hence find the probability of a Type I error for this test. [2]
4. (d) The observed value is 10. State the conclusion of the test, in context. [2]
5. (e) Describe a Type II error in this context. [1]
6. (f) In fact the probability is 0.3. Find the probability of a Type II error. [3]

A harbour has averaged 0.8 boats arriving late per day. After the timetable is changed, 13 boats arrive late in the next 10 days. The harbour master believes that the mean number of late boats per day has increased, and tests his belief at the 5% significance level. Assume that late arrivals occur at random. [10]

1. (a) Write down suitable null and alternative hypotheses, defining the parameter you use. [2]
2. (b) Write down the distribution of \(X\), the number of late boats in 10 days, when \(H_0\) is true. [1]
3. (c) Find the \(p\)-value for the test and state the conclusion at the 5% significance level, in context. [4]
4. (d) Find the critical region for the test at the 5% significance level, for a count over 10 days. [2]
5. (e) Find the probability of a Type I error for this test. [1]

Faults occur in a pipeline at random, at a mean rate of 0.4 per kilometre. After a repair programme, the engineer believes that the rate has increased. He counts the number of faults in a 5 km section, \(Y\), and will test his belief at the 5% significance level. [10]

1. (a) Write down suitable null and alternative hypotheses, defining the parameter you use. [2]
2. (b) Write down the distribution of \(Y\), the number of faults in 5 kilometres, when \(H_0\) is true. [1]
3. (c) Find the critical region for the test at the 5% significance level. [2]
4. (d) Write down the actual significance level of the test. [1]
5. (e) Find the probability of a Type II error when the true rate is 0.9 per kilometre. [2]
6. (f) Another engineer counts the faults in a 3 km section, \(Z\), to test whether the rate has decreased below 0.4 per kilometre at the 5% significance level. Show that this test has no critical region, and find the smallest significance level for which a critical region exists. [2]

A researcher records the height, \(x\) cm, and the reaction time, \(y\) milliseconds, of eight adults. She wishes to test whether there is any correlation between height and reaction time. [7]

1. (a) State the null hypothesis and the alternative hypothesis, defining the symbol you use. [2]
2. (b) Use your calculator to find the product moment correlation coefficient \(r\) and the \(p\)-value for the test. [2]
3. (c) Conclude at the 5% significance level, in context. [2]
4. (d) State an assumption that is needed for the test to be valid. [1]

The time, \(X\) seconds, that a robot takes to complete one task is normally distributed with mean 120 and standard deviation 8. [13]

1. (a) Find the probability that one task has a time less than 110 s. [1]
2. (b) Two tasks are chosen at random. Find the probability that their total time is less than 230 s. [3]
3. (c) The time is thought to have decreased. A random sample of 36 tasks is used to test this at the 5% significance level, using the sample mean \(\bar X\) and the known standard deviation \(8\). Write down the null hypothesis and the alternative hypothesis. [1]
4. (d) Find the critical region for \(\bar X\). [3]
5. (e) The test is carried out after a change in the process. Every task now has its time decreased by 3%. Find the new mean and the new standard deviation of the time. [2]
6. (f) Use these values to find the probability of a Type II error. [3]