Question

4. Columns is a video game available on the Sega Gen- esis system. In the flashing version of the game, the player attempts to capture a flashing block in the bottom row of the screen as quickly as possible. The game may be played at levels 2 through 9. The level denotes the number of complete rows of blocks on the screen at the beginning of the game. For example at level 2 the player must break through the second row of blocks in order to reach the flashing block in the first row. At level 9, the player must break through eight rows of blocks in order to reach the flashing block. I performed a CRD with five games (trials) at each of the eight levels. The response Y is the time needed to capture the flashing block, and the predictor X is the level. Before I collected any data, it seemed obvious, based on my prior experience 5· 7 playing the game, that increasing the level makes the 6 game more difficult. Thus, I anticipated finding an increasing relationship between Y and X. I further an Exercises 13.2 459 ticipated three sources of variation that would cloud the relationship between Y and X. First, although a game at level 5, for example, always begins with 5 rows of blocks, the game's assignment of the color to the blocks can affect the difficulty of my task. Sec- ond, the colors of new blocks (which either help me capture blocks or get in the way) given to me by the computer can affect the difficulty of the game. Fi- nally, my decisions on where to put the new blocks presumably vary from game to game. I believed that the higher the level, the greater would be the varia- bility in the responses. In summary, I suspected that as the level increased, both the response and its vari- ability would increase. Figure 13.13 presents the scatterplot of the data from the CRD. Use this plot to answer the following: (a) Do you think there are any isolated cases? If yes, (b) One of the numbers below is the correlation coef- which one(s)? ficient for these data; which one is it? 1.00,-0.524,0.026,0.315,0.716 (c) Table 13.5 presents summary statistics for the responses for each of the eight levels. Discuss how these numbers support or fail to support my conjectures that the response and its variability should increase as the level increases. 3 / Correlation and Regression Time (seconds) 200 180 160 1I 140 120 100 80 60 40 20 Level FIGURE 13.13 Scatterplot of time to finish versus level for 40 plays of the Columns video game Selected Summary Statistics of Response by Level for 40 Games of TABLE 13.5 Columns Level Mean 3 4 5 6 7 8 9 2 13.8 19.2 42.2 78.4 60.0 75.2 103.0 106.2 10.0 10.0 40.0 69.0 58.0 63.0 80.0 116.0 Standard Deviation 11.0 14.0 23.4 40.3 7.4 35.2 59.8 30.8 15.5 26.0 46.5 75.5 14.0 53.5 82.5 58.5 Median IQR

Answer 1

(a) case(8,250), the upper-most observation may treated as isolated case.

(b) correlation coefficient should be 0.716,

as the points on the scattered plot is concentrated from quardent III to quardent I. which means the positive correlation. also points are more concentrated diagonally which means strong correlation may be more than( r=) 0.5

(c) data supports that response and variability increases as level increases.

since there is increasing trend in mean and median (response) also there is increasing trend in standard deviation (variability) as level increases