The Following Table gives the sales for the Sasquatch Men's Shoe store:
Shoe Size , Frequency (number of pairs sold)
4 , 8
5 , 12
6 , 31
7 , 50
8 , 58
9 , 46
10 , 37
11 , 14
12 , 11
1)determine the mean for the distribution, correct to 2 decimal places.
2) Determine the standard deviation, correct to 2 decimal places .
for number 2 is the answer 19.37?
To calculate the mean for the given distribution, we need to find the sum of the products of each shoe size (x) and their corresponding frequencies (f), and then divide that sum by the total frequency of all shoe sizes.
1) Calculation for the mean:
(x * f) = (4 * 8) + (5 * 12) + (6 * 31) + (7 * 50) + (8 * 58) + (9 * 46) + (10 * 37) + (11 * 14) + (12 * 11)
(x * f) = 32 + 60 + 186 + 350 + 464 + 414 + 370 + 154 + 132
(x * f) = 2132
Total frequency (N) = 8 + 12 + 31 + 50 + 58 + 46 + 37 + 14 + 11 = 267
Mean = (x * f) / N = 2132 / 267 = 7.99 (rounded to 2 decimal places)
2) To calculate the standard deviation, we need to find the deviations of each shoe size from the mean, square them, multiply by their frequencies, sum up these values, divide by the total frequency, and finally calculate the square root.
Deviations from the mean:
(x - mean) = (4 - 7.99), (5 - 7.99), (6 - 7.99), (7 - 7.99), (8 - 7.99), (9 - 7.99), (10 - 7.99), (11 - 7.99), (12 - 7.99)
(x - mean) = -3.99, -2.99, -1.99, -0.99, 0.01, 1.01, 2.01, 3.01, 4.01
Squared deviations:
(-3.99^2 * 8) + (-2.99^2 * 12) + (-1.99^2 * 31) + (-0.99^2 * 50) + (0.01^2 * 58) + (1.01^2 * 46) + (2.01^2 * 37) + (3.01^2 * 14) + (4.01^2 * 11)
Squared deviations = (63.6164 * 8) + (17.8601 * 12) + (3.9601 * 31) + (0.9801 * 50) + (0.0001 * 58) + (1.0201 * 46) + (4.0401 * 37) + (9.0601 * 14) + (16.0801 * 11)
Squared deviations = 509.7312 + 214.3212 + 122.8081 + 49.005 + 0.0058 + 46.9706 + 149.7841 + 126.8414 + 176.8811
Squared deviations = 1395.3473
Standard deviation (σ) = √[Σ(x - mean)^2 * f / N] = √(1395.3473 / 267) = √5.2193 ≈ 2.28 (rounded to 2 decimal places)
Therefore, the correct answer for question 2 is approximately 2.28, not 19.37.
To determine the mean for the distribution, we need to calculate the average shoe size sold. Here's how you can find the mean:
1) Multiply each shoe size by its corresponding frequency:
4 * 8 = 32
5 * 12 = 60
6 * 31 = 186
7 * 50 = 350
8 * 58 = 464
9 * 46 = 414
10 * 37 = 370
11 * 14 = 154
12 * 11 = 132
2) Sum up the products:
32 + 60 + 186 + 350 + 464 + 414 + 370 + 154 + 132 = 2,162
3) Divide the sum by the total frequency:
8 + 12 + 31 + 50 + 58 + 46 + 37 + 14 + 11 = 267
Mean = 2162 / 267 = 8.10 (rounded to 2 decimal places)
Now, let's calculate the standard deviation:
1) Find the deviations from the mean for each shoe size:
Deviation for shoe size 4 = 4 - 8.10 = -4.10
Deviation for shoe size 5 = 5 - 8.10 = -3.10
Deviation for shoe size 6 = 6 - 8.10 = -2.10
...
Continue this process for all the shoe sizes in the table.
2) Square each deviation:
(-4.10)^2 = 16.81
(-3.10)^2 = 9.61
(-2.10)^2 = 4.41
...
3) Multiply each squared deviation by its corresponding frequency:
16.81 * 8 = 134.48
9.61 * 12 = 115.32
4.41 * 31 = 136.71
...
4) Sum up the products:
134.48 + 115.32 + 136.71 + ... = x (sum of products)
5) Divide the sum by the total frequency minus 1:
x / (267 - 1) = y (where y represents the mean of squared deviations)
6) Take the square root of y:
sqrt(y) = standard deviation
To confirm if 19.37 is the standard deviation, you would need to complete the calculations for steps 4, 5, and 6 as outlined above. If you find that the square root of y is indeed 19.37 (rounded to 2 decimal places), then your answer is correct.