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]