Four different brands of 18 ounce peanut butter were randomly selected from a grocer's shelves; their contents were weighed. Use these results to determine which company fills its jars more consistently?
Answer
Bama: 18.11, 17.98, 18.03, 17.99, 17.95
Skippy: 18.03, 17.96, 17.89, 17.95,17.98
Jiff: 18.02, 18.01, 17.99, 18.01, 17.99
Hytop: 17.95, 18.05, 17.98, 18.01, 18.02
Bama: 18.012 Average; + 0.012
Skippy: 17.962 Average; - 0.338
Jiff: 18.004 Average; + 0.004
Hytop: 18.002 Average; + 0.002
While Hytop's average is closest to 18, Jiff's weights are all within 0.1% of the advertised weight.
To determine which company fills its jars more consistently, you can calculate the standard deviation for each brand. The standard deviation will give you a measure of how spread out the weights are for each brand. A smaller standard deviation indicates that the weights are more consistent.
First, let's calculate the mean weight for each brand by adding up all the weights and dividing by the number of weights:
Bama: (18.11 + 17.98 + 18.03 + 17.99 + 17.95) ÷ 5 = 17.99
Skippy: (18.03 + 17.96 + 17.89 + 17.95 + 17.98) ÷ 5 = 17.96
Jiff: (18.02 + 18.01 + 17.99 + 18.01 + 17.99) ÷ 5 = 18.004
Hytop: (17.95 + 18.05 + 17.98 + 18.01 + 18.02) ÷ 5 = 18.002
Next, subtract each individual weight from the mean for each brand and square the result:
Bama: (18.11 - 17.99)^2, (17.98 - 17.99)^2, (18.03 - 17.99)^2, (17.99 - 17.99)^2, (17.95 - 17.99)^2
Skippy: (18.03 - 17.96)^2, (17.96 - 17.96)^2, (17.89 - 17.96)^2, (17.95 - 17.96)^2, (17.98 - 17.96)^2
Jiff: (18.02 - 18.004)^2, (18.01 - 18.004)^2, (17.99 - 18.004)^2, (18.01 - 18.004)^2, (17.99 - 18.004)^2
Hytop: (17.95 - 18.002)^2, (18.05 - 18.002)^2, (17.98 - 18.002)^2, (18.01 - 18.002)^2, (18.02 - 18.002)^2
Now, sum up all of the squared differences for each brand:
Bama: sum of squared differences = 0.025
Skippy: sum of squared differences = 0.0295
Jiff: sum of squared differences = 0.002
Hytop: sum of squared differences = 0.025
Then, divide the sum of squared differences by the number of weights minus 1, and take the square root of the result to get the standard deviation:
Bama: square root(0.025 ÷ (5 - 1)) = 0.0884
Skippy: square root(0.0295 ÷ (5 - 1)) = 0.121
Jiff: square root(0.002 ÷ (5 - 1)) = 0.025
Hytop: square root(0.025 ÷ (5 - 1)) = 0.0884
Comparing the standard deviations, we can see that Jiff brand has the smallest value of 0.025. Therefore, Jiff fills its jars more consistently compared to the other brands.