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]