Question 4. [ 21 marks ] [Second half of Chapter 8] A Sudoku puzzle is completed using given numbers, letters or symbols to fill in a grid. Two statistics lecturers decided to investigate whether the characters used in the puzzle affected the time taken to complete the puzzle. The 227 participants in the study were all students in an introductory statistics class at an Irish university. Usually Sudoku puzzles are completed on a 9 by 9 grid. To reduce the time needed to complete the puzzles, a 6 by 6 Sudoku grid was used instead. All puzzles had the same layout but four different types of characters were used (Greek, Letter, Number, Symbol ) as shown in the table below: Sudoku type Characters Greek A Letter Number 3 5 6 Symbol A X Students were randomly allocated to one of the four Sudoku types. Puzzles were completed under test conditions. A clock was displayed which students used to record the time (in seconds) taken to correctly complete their puzzle. Note: Use the computer output from SPSS given to the right to answer (e) to (g). (a) For this study describe the: [ 1 mark ] (i) units, (ii) treatment or factor of interest, (iii) response. (b) (i) Use iNZight to draw the appropriate plots for this data set. [ 1 mark ] (ii) Comment on any features in the plots in terms of the original story. [ 3 marks ] (c) Use iNZight to get summary statistics for this data and calculate the ratio of the largest sample standard deviation to the smallest sample standard deviation. (Note: The ratio of A to B is A divided by B.) [ 1 mark ] Note: The data file SudokuData . cav is available on Canvas under Assignments > Assignment 3. (d) Comment on the validity of the F-test by briefly discussing each assumption in terms of the original story. [ 4 marks ] (e) Assume that an F-test is an appropriate test to use here. (Note: It may not be.) (i) State the null hypothesis for the test, both in words and using symbols. [ 1 mark ] (ii) State the alternative hypothesis for the test in words. [ 1 mark ] (iii) What does the result of the F-test tell you about the underlying mean time taken to correctly complete the puzzle for the four Sudoku types? [ 1 mark ](f) Assume the Tukey's pairwise comparisons are valid and appropriate. (Note: They may not be.) (i) Investigate whether, on average, there is a difference between the time taken to correctly complete the puzzle when the Sudoku type is a Number character and when the Sudoku type is a Greek character. Interpret the P-value and confidence interval. Note: A conclusion is not required here. [ 2 marks ] (ii) Between which pair (or pairs) of Sudoku types were there significant differences (at the 5% level) in the mean time taken to correctly complete the puzzle? [ 1 mark ] (iii) Are we able to determine which Sudoku type has the fastest (i.e., lowest) underlying mean time taken to correctly complete the puzzle? If so, name the character type. [ 1 mark ] (g) In three to five sentences, provide an overall conclusion for this study. [ 4 marks ] Computer output from SPSS for use in Question 4 (e) to (g) ANOVA Time (seconds) Sum of Squares Mean Square F Sig Between Groups 389465.706 3 129821.902 16.032 .000 Within Groups 1805799.712 223 8097-757 Total 2195265.419 226 Multiple Comparisons Dependent Variable: Time (seconds) Tukey HSD Mean 95% Confidence Interval (I) Sudoku type (J) Sudoku type Difference (I-J) Sid. Error Sig Lower Bound |Upper Bound Greek Letter 66.182 17.186 101 21.70 1 10.67 Number 110.695 16.985 66.73 154.66 Symbol 29.668 17.330 320 -15.19 74.53 Letter Greek -66.182" 17.186 001 -110.67 -21.70 Number 44.512 16.504 138 1.79 87.23 Symbol -36.515 16.859 136 -80.15 7.12 Number Greek -110.695 16.985 000 -154.66 -66.73 Letter -44.512 16.504 138 -87.23 -1.79 Symbol -81.027 16.654 -124.14 -37.92 Symbol Greek -29.668 17.330 320 -74.53 15.19 Letter 36.515 16.859 136 -7.12 80.15 Number 81.027 16.654 000 37.92 124.14 *. The mean difference is significant at the 0.05 level