An adverting firm wishes to demonstrate to its clients the effectiveness of the advertising campaigns it has conducted. Using x as cost in millions of dollars: 3.5o, 3.05, 3.64, 1.59, 3.21, 2.12, 2.07, 3.99,1.58, 3.01, 1.36, 2.24; y for increases in sales: 6.89, 6.62, 6.73, 6.65, 6.49, 6.49, 6.81, 6.99, 6.29, 6.98, 6.42,6.63; and xy:24.115, 20.191, 24.4972, 10.5735, 20.8329, 13.7588, 14.0967, 27,8901, 9.9382, 21.0098, 8.7312, 14.8512. I have 12 recent campaigns to use for n and added all of x and divided by 12 to get 2.6133; and all of y's divided by 12 to get 6.6658; my equation looks like this 210.4856-(12)(2.36133)(6.6658)/(12-1) as (n-1), but I do not know how to get my sx or my sy (sx is my sample standard deviation of x values and y is my sample standard deviation of the y values). Can you help tell me how to get my sx and my sy sample standard deviations?
To calculate the sample standard deviation (sx and sy) for the given x and y values, you need to follow these steps:
1. Calculate the mean of the x-values (2.6133) and the mean of the y-values (6.6658).
2. Calculate the difference between each individual x-value and the mean of x-values. Square each difference.
3. Sum up all the squared differences from step 2.
4. Divide the sum from step 3 by (n-1), where n is the number of observations (12 in this case). This gives you the variance of x-values (sx^2).
5. Take the square root of the variance obtained in step 4 to get the sample standard deviation of x-values (sx).
6. Repeat steps 2-5 for the y-values to obtain the sample standard deviation of y-values (sy).
Let's calculate sx and sy using the given data:
1. Calculate the mean of x-values:
Mean of x = (3.5 + 3.05 + 3.64 + 1.59 + 3.21 + 2.12 + 2.07 + 3.99 + 1.58 + 3.01 + 1.36 + 2.24) / 12 = 2.6133
Calculate the mean of y-values:
Mean of y = (6.89 + 6.62 + 6.73 + 6.65 + 6.49 + 6.49 + 6.81 + 6.99 + 6.29 + 6.98 + 6.42 + 6.63) / 12 = 6.6658
2. Calculate the differences between each x-value and the mean of x-values, and then square each difference:
(3.5 - 2.6133)^2, (3.05 - 2.6133)^2, (3.64 - 2.6133)^2, (1.59 - 2.6133)^2, (3.21 - 2.6133)^2, (2.12 - 2.6133)^2, (2.07 - 2.6133)^2, (3.99 - 2.6133)^2, (1.58 - 2.6133)^2, (3.01 - 2.6133)^2, (1.36 - 2.6133)^2, (2.24 - 2.6133)^2
3. Sum up all the squared differences:
Sum of squared differences of x = (3.5 - 2.6133)^2 + (3.05 - 2.6133)^2 + (3.64 - 2.6133)^2 + (1.59 - 2.6133)^2 + (3.21 - 2.6133)^2 + (2.12 - 2.6133)^2 + (2.07 - 2.6133)^2 + (3.99 - 2.6133)^2 + (1.58 - 2.6133)^2 + (3.01 - 2.6133)^2 + (1.36 - 2.6133)^2 + (2.24 - 2.6133)^2
4. Divide the sum of squared differences by (n-1) to obtain the variance of x-values (sx^2):
Variance of x = Sum of squared differences of x / (n-1)
5. Take the square root of the variance of x to obtain the sample standard deviation of x-values (sx):
sx = √(Variance of x)
6. Repeat steps 2-5 for the y-values to obtain the sample standard deviation of y-values (sy).