Questions 1-6 whole sheetQuestion 1 The lifetimes (in units of 106 seconds) of certain satellite components are shown in the frequency distribution given in ‘Dataset1’. 1. Draw a frequency polygon,...

Question 1 The lifetimes (in units of 106 seconds) of certain satellite components are shown in the frequency distribution given in ‘Dataset1’. 1. Draw a frequency polygon, histogram and cumulative frequency polygon for the data. 2. Calculate the frequency mean, the frequency standard deviation, the median and the first and third quartiles for this grouped data. 3. Compare the median and the mean and state what this indicates about the distribution. Comment on how the answer to this question relates to your frequency polygon and histogram. 4. Explain the logic behind the equations for the mean and standard deviation for grouped data, starting from the original equations for a simple list of data values. (This does not just mean ’explain how the equations are used’.) Page 2 5. Carry out an appropriate statistical test to determine whether the data is normally distributed. Question 2 A manufacturer of metal plates makes two claims concerning the thickness of the plates they produce. They are stated here: • Statement A: The mean is 200mm • Statement B: The variance is 1.5mm2 . To investigate Statement A, the thickness of a sample of metal plates produced in a given shift was measured. The values found are listed in Part (a) of worksheet ‘Dataset2’, with millimetres (mm) as unit. 1. Calculate the sample mean and sample standard deviation for the data in Part (a) of ’Dataset2’. Explain why we are using the phrase ’sample’ mean or sample’ standard deviation. 2. Set up the framework of an appropriate statistical test on Statement A. Explain how knowing the sample mean before carrying out the test will influence the structure of your test. 3. Carry out the statistical test and state your conclusions. To investigate the second claim, the thickness of a second sample of metal sheets was measured. The values found are listed in Part (b) of worksheet ‘Dataset2’, with millimetres (mm) as unit. 1. Calculate the sample mean and then the sample variance and standard deviation for the data in Part (b). Page 3 2. Set up the framework of an appropriate statistical test on Statement B. Explain how knowing the sample variance before carrying out the test would influence the structure of your test. 3. Carry out the statistical test and state your conclusions. Question 3 A manager of an inter-county hurling team is concerned that his team lose matches because they ‘fade away’ in the last ten minutes. He has measured GPS data showing how much ground particular players cover within a given time period; this is the data in list (a) in worksheet ‘Dataset3’. He has acquired the corresponding data from an opposing, more successful team, which is given in list (b). 1. Calculate the sample mean and sample standard deviation for the two sets of data. 2. Set up the frame work of an appropriate statistical test to determine whether there is a difference in the distances covered by the two groups of players. 3. Explain how having the results of the calculations above in advance of doing your statistical test will influence the structure of that test. 4. Carry out the statistical test and state your conclusions. Question 4 A study was carried out to determine whether the resistance of the control circuits in a machine are lower when the machine motor is Page 4 running. To investigate this question, a set of the control circuits was tested as follows. Their resistance was measured while the machine motor was not running for a certain period of time and then again while the motor was running. The values found are listed in worksheet ‘Dataset4’, with kilo-Ohms as the unit of measurement. 1. Set up the structure of an appropriate statistical test to determine whether the resistance of the control circuit in a machine are lower when the machine motor is running. 2. Explain how the order of subtraction chosen to calculate the differences will influence the structure of the test. 3. Give a reason why the data is measured with the engine not running first and then with the engine running. 4. Explain how knowing the mean of the differences in advance will influence the structure of your statistical test. 5. Carry out the statistical test and state your conclusions. Question 5 A study was carried out to determine the influence of a trace element found in soil on the yield of potato plants grown in that soil, defined as the weight of potatoes produced at the end of the season. A large field was divided up into 14 smaller sections for this experiment. For each section, the experimenter recorded the amount of the trace element found (in milligrams per metre squared) and the corresponding weight of the potatoes produced (in kilograms). This information is presented in the worksheet ‘Dataset5’ in the Excel document. Define X as the trace element amount and Y as the yield. Page 5 1. Draw a scatterplot of your data set. 2. Calculate the coefficients of a linear equation to predict the yield Y as a function of X. 3. Calculate the correlation coefficient for the paired data values. 4. Set up the framework for an appropriate statistical test to establish if there is a correlation between the amount of the trace element and the yield. Explain how having the scatterplot referred to above and having the value of r in advance will influence the structure of your statistical test. 5. Carry out and state the conclusion of your test on the correlation. 6. Comment on how well the regression equation will perform based on the results above. Question 6 A multinational corporation is conducting a study to see how its employees in five different countries respond to three gifts in an incentive scheme. The numbers of employees who choose each of the three gifts (G1 to G3) in each of the five countries (A to E) are given in the table in ‘Dataset6’ in the Excel document. 1. Set up the structure of an appropriate statistical test to determine whether the data supports a link between choice of gift and country, including the statistic to be used. 2. Carry out this test, showing clearly in your work how the expected values are calculated for your test statistic.