Hypothesis testing: exam review Exam review

A cereal company states that the mean mass of its boxes is 500 g. An inspector believes that the mean mass is less than this. She will test her 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) She weighs a random sample of 25 boxes and finds a sample mean of 492 g. If the company's statement is true, the probability of a sample mean of 492 g or less is 0.004. State whether the inspector has sufficient evidence to support her belief, and give a reason. [3]

A café states that its customers choose espresso, latte, cappuccino or another drink in the proportions 0.4, 0.3, 0.2 and 0.1. The café records the drink chosen by each of 200 customers. The results are in the table. A goodness of fit test is used to test the café's statement. [8]

1. (a) State the null hypothesis and the alternative hypothesis. [1]
2. (b) Find the expected frequency for each drink. [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 six-sided die is rolled 120 times. A goodness of fit test of the null hypothesis that the die is fair gives \(\chi^2=14.6\). [6]

1. (a) Write down the number of degrees of freedom. [1]
2. (b) Find the critical value of the test at the 5% 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 manufacturer states that the mean mass of its bags of sugar is 1000 g. A trading standards officer believes that the mean mass is less than this. She weighs 10 randomly chosen bags. The masses, in grams, are 1002, 1003, 996, 992, 991, 996, 991, 989, 997, 997. [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]

Two machines fill tubs of margarine. A random sample of 12 tubs from machine 1 has a mean mass of 251.8 g and \(s_{n-1}=3.1\) g. A random sample of 15 tubs from machine 2 has a mean mass of 248.2 g and \(s_{n-1}=3.4\) g. A test is carried out to see whether the mean masses of the tubs from the two machines 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 judges each rank eight dancers, A to H, from best to worst. Judge 1 orders the dancers C, A, F, B, H, D, E, G. Judge 2 orders them A, C, F, H, D, B, E, G. A one-tailed test is used, at the 5% significance level, to see whether the judges tend to agree. The critical value is 0.643. [6]

1. (a) Write down the rank that each judge gave to each dancer, for dancers 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 battery maker states that each battery is faulty with probability 0.15, independently of the others. An inspector tests 160 packs of 4 batteries and records \(X\), the number of faulty batteries in each pack. The results are in the table. A goodness of fit test is used to test whether \(X\sim B(4, 0.15)\). [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 \(X=3\) and \(X=4\) cannot be kept as separate groups. [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 166 students in Year 10 and Year 11 which type of music they prefer. The results are in the table. A \(\chi^2\) test for independence is used to test whether music preference is independent of year group. [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 Year 11 and Rock. [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]

Ten students record the time they spend on social media each day, \(x\) hours, and the time they sleep that night, \(y\) hours. The values of \(x\) are 1.5, 2.0, 2.5, 3.0, 3.0, 3.5, 4.0, 4.5, 5.0, 7.5 and the values of \(y\) are 8.5, 8.0, 8.2, 7.5, 7.0, 7.6, 6.5, 6.8, 6.0, 5.5. A test is carried out, at the 5% significance level, to see whether more time on social media goes with less sleep. The critical value for this one-tailed test is 0.564. [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 \(x\), 7.5, was recorded wrongly. The correct value is 6.0. Without calculating, state how this changes \(r_s\), and give a reason. [2]
7. (g) The values of \(y\) are converted from hours to minutes. State how this changes \(r_s\). [1]

A shop asks 194 customers how they paid and which age group they are in. A \(\chi^2\) test for independence is to be used to test whether the way a customer pays is independent of age group. [10]

1. (a) Show how to find the expected frequency for Card and 65 or over. [2]
2. (b) Write down one expected frequency that is less than 5, and explain why 45 to 64 and 65 or over 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 city bus company records the number of breakdowns in each of 100 weeks. The results are in the table. The company thinks that the number of breakdowns in a week 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 breakdowns in a week. [2]
2. (b) Using this mean, show that the expected frequency of 2 breakdowns is 25.2, correct to 3 significant figures. [2]
3. (c) The expected frequency of exactly 4 breakdowns is 4.79. 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 bottling plant states that the mean volume of a bottle of juice is 330 ml. An inspector believes that the mean volume is less than this. The volume of juice in a bottle is normally distributed with standard deviation 2.5 ml, which is known. A random sample of 40 bottles has a mean volume of 329.2 ml. [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 inspector then tests a second production line, where the standard deviation is not known. The volumes, in ml, of a random sample of 10 bottles from this line are: 328.1, 330.4, 327.9, 329.5, 328.8, 330.1, 327.6, 329.0, 328.4, 329.7. Assuming that these volumes are normally distributed, test the same belief for this line. Find the \(p\)-value and state the conclusion at the 5% significance level, in context. [3]

A gardener compares two fertilisers, X and Y. She grows 30 plants with fertiliser X and 35 different plants with fertiliser Y. The heights of the plants are normally distributed. The population standard deviations are known to be 5.1 cm for X and 6.0 cm for Y. The mean height of the plants with fertiliser X is 41.2 cm and the mean height of the plants with fertiliser Y is 44.0 cm. She believes that fertiliser Y gives taller plants. [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]

Eight athletes take part in a six-week training programme. Each athlete's resting heart rate, in beats per minute, is measured before and after the programme. A coach believes that the programme reduces the mean resting heart rate. [8]

1. (a) State the null hypothesis and the alternative hypothesis, taking \(d\) as the after value minus the before 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 company compares two methods, A and B, of assembling a unit. A random sample of 40 units assembled by method A has a mean assembly time of 31.2 minutes. An independent random sample of 45 units assembled by method B has a mean assembly time of 27.9 minutes. The population standard deviations are known to be 4.8 minutes for method A and 5.1 minutes for method B. The distributions of the assembly times are not known. The company claims that method A takes more than 2 minutes longer, on average, than method B. Let \(\mu_A\) and \(\mu_B\) be the population mean assembly times for methods 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 fills tubs with soup. The mass of soup in a tub, \(X\) g, is normally distributed with standard deviation 6 g, and the mass is claimed to have mean 400 g. An inspector takes a random sample of 36 tubs and tests, at the 5% significance level, whether the mean mass differs from 400 g. Let \(\mu\) be the population mean mass. [9]

1. (a) Write down the distribution of \(\bar X\), the mean of the sample of 36, 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 402.3 g. State the conclusion of the test, in context. [1]
4. (d) A second inspector rejects \(H_0\) if \(\bar X\) differs from 400 by more than 1.5 g. Find the probability that this inspector makes a Type I error. [2]
5. (e) The true value of \(\mu\) is in fact 402. Using the critical region in part (b), find the probability of a Type II error. [2]

A seed supplier states that 70% of its seeds germinate. A gardener believes that the proportion is lower. She sows 25 seeds, chosen at random, and counts the number \(X\) that germinate. 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 12. 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 0.55. Find the probability of a Type II error. [3]

A weaving mill has averaged 3.5 flaws per week in the cloth it produces. After new looms are fitted, the manager records 7 flaws in the next 4 weeks. She believes that the mean number of flaws per week has decreased, and tests her belief at the 5% significance level. Assume that flaws 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 flaws in 4 weeks, 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 4 weeks. [2]
5. (e) Find the probability of a Type I error for this test. [1]

Customers arrive at a help desk at random, at a mean rate of 0.5 per hour. After a change to the opening hours, the manager believes that the rate has increased. She counts the number of customers who arrive in a 6-hour period, \(Y\), and will test her 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 customers arriving in 6 hours, 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 1 per hour. [2]
6. (f) Another manager counts the customers arriving in a 4-hour period, \(Z\), to test whether the rate has decreased below 0.5 per hour 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 teacher records the number of hours, \(x\), that nine students spent revising for a test and their score, \(y\), out of 80. She believes that there is positive correlation between the time spent revising and the score. [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 mass, \(X\) grams, of a bag of rice packed by a machine is normally distributed with mean 500 and standard deviation 6. [13]

1. (a) Find the probability that one bag has a mass greater than 508 g. [1]
2. (b) Two bags are chosen at random. Find the probability that their total mass is greater than 1012 g. [3]
3. (c) The mass is thought to have increased. A random sample of 16 bags is used to test this at the 5% significance level, using the sample mean \(\bar X\) and the known standard deviation \(6\). 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 bag now has its mass increased by 0.8%. Find the new mean and the new standard deviation of the mass. [2]
6. (f) Use these values to find the probability of a Type II error. [3]