An experiment was conducted at a small supermarket for a period of 8 days on the sales of a single brand of dog food, involving three levels of shelf height: knee level, waist level, and eye level. During each day, the shelf height of the dog food was randomly changed on three different occasions. Sales, in hundreds of dollars, of the dog food per day for the three shelf heights are given below. Based on the data, is there a significant difference in the average daily sales of this dog food based on shelf height? Use a 0.05 level of significance. Shelf Height Knee Level Waist Level Eye Level 77 88 85 82 94 85 86 93 87 78 90 81 81 91 80 86 94 79 77 90 87 81 87 93

Question

Brogan Oliver

Mathematics

11 July, 11:45

An experiment was conducted at a small supermarket for a period of 8 days on the sales of a single brand of dog food, involving three levels of shelf height: knee level, waist level, and eye level. During each day, the shelf height of the dog food was randomly changed on three different occasions. Sales, in hundreds of dollars, of the dog food per day for the three shelf heights are given below. Based on the data, is there a significant difference in the average daily sales of this dog food based on shelf height? Use a 0.05 level of significance. Shelf Height Knee Level Waist Level Eye Level 77 88 85 82 94 85 86 93 87 78 90 81 81 91 80 86 94 79 77 90 87 81 87 93

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Derek Glass · Answer 1

There is no significant difference in the average daily sales of this dog food based on shelf height

Step-by-step explanation:

Null hypothesis: There is no significant difference in the average sales

Alternate hypothesis: There is a significant difference in the average daily sales

Using the F-table, at 0.05 significance level and (2, 21) degrees of freedom (3-1, 24-3), the critical value is 3.47

Conclusion: Reject the null hypothesis if the computed F exceeds 3.47

Data values

Knee level: 77,88,85,82,94,85,86,93 (Mean is 86.25)

Waist level: 87,78,90,81,81,91,80,86 (Mean is 84.25)

Eye level: 94,79,77,90,87,81,87,93 (Mean is 86)

Grand mean = (86.25+84.25+86) / 3 = 85.5

Sum of Squares (SS) for knee level = summation (sales - grand mean) ^2 = 220

SS for waist level = 180

SS for eye level = 288

SStotal = 220+180+288 = 688

Sum of Squares due to Treatment (SST) for knee level = n (mean - grand mean) ^2 = 8 (86.25 - 85.5) ^2 = 4.5

SST for waist level = 12.5

SST for eye level = 0.25

Total SST = 4.5+12.5+0.25 =

17.25

Mean Sum of Treatment (MST) = SST / (number of food types - 1) = 17.25 / (3-1) = 17.25/2 = 8.625

Sum of Squares due to Error (SSE) = SStotal - SST = 688 - 17.25 = 670.75

Mean Sum of Error (MSE) = SSE / (24-3) = 670.75/21 = 31.940

F = MST/MSE = 8.625/31.940 = 0.27

The computed F 0.27 is less than the critical value 3.47, so we fail to reject the null hypothesis

Conclusion: There is no significant difference in the average daily sales of this dog food based on shelf height