Question

G R

57 62

46 58

85 81

80 88

95 84

31 54

56 44

40 65

52 37

26 51

93 76

54 43

67 64

42 59

29 51

81 70

35 49

59 61

44 57

84 97

34 55

49 44

73 86

74 89

44 52

41 49

72 61

60 48

48 69

92 87

64 77

52 47

58 66

84 80

60 50

49 38

96 74

20 49

42 19

36 58

69 48

44 56

37 59

57 29

31 62

74 51

85 79

19 52

33 76

48 80

88 84

72 64

45 58

36 42

64 85

77 75

28 22

93 87

45 48

50 40

A private-label bottler of soft drinks asks each of 60 members of a tasting panel (who are regarded as a random sample from millions of potential customers) to rate each of two possible formulations of a cola drink on a 100-point scale; higher scores are desirable. Formulation G is less expensive and will be used unless there is a clear evidence that formulation R is preferred. From the data, the bottler obtains the difference (R-G) in the ratings for each panelist. After calculating and examining the 90% and the 95% confidence intervals for the mean of the difference in the ratings, the management of the private-label bottler concluded that the difference in the ratings is negligible. Consequently, they have decided to use formulation G which is less expensive. The data on the panelist’s ratings they have used is in the Excel data file named ‘Cola Ratings’. Using the same data, please re-calculate the confidence intervals you think the management’s decision must have been based on. Show all the necessary steps and interpret your confidence intervals. Based on your results, do you agree with the management’s conclusion? Please justify your answer.

Answer 1

First, we have to compute the mean of the difference and standard deviation of the difference. From the data, it can be computed:

μ_d = 4.117

s_d = 16.548

The confidence interval formula is:

Here, n = 60

For 90% confidence level, t* (df = 59) = 1.671

Hence, Confidence interval:

4.117 - 3.570 < d < 4.117 + 3.570

Confidence Interval: 0.547 < d < 7.687

The confidence interval at 90% confidence level says that there must be a difference between the ratings as it does not include 0.

For 95% confidence level, t* (df = 59) = 2.001

Hence, confidence interval:

4.117 - 4.275 < d < 4.117 + 4.275

Confidence Interval: -0.158 < d < 8.392

Now, as the management tried to increase their confidence level to 95%, the interval includes 0 and hence it can be concluded that there is no significant difference between the ratings of the two formulations. Hence, it would be a wise choice to go for formulation G as it is less expensive and as the consumers don't perceive any difference in the ratings between G and R.