To study the efficiency of its new price-scanning equipment, a local supermarket monitored the amount of time its customers had to wait in line. The frequency distribution in the following table summarizes the findings. Find the standard deviation of the amount of time spent in line. (Round your answer to three decimal places.)
______Min
x = Time (minutes) Number of Customers
0 ≤ x < 1 75
1 ≤ x < 2 58
2 ≤ x < 3 64
3 ≤ x < 4 40
4 ≤ x < 5 38
I tried
Frequency Midpoint (xm) f*xm (f*xm)^2
75 .5 37.5 140625
58 1.5 87 7569
64 2.5 160 25600
40 3.5 140 19600
38 4.5 171 29241
+------ +------ +-------
275 595.5 83416.25
(f*xm)^2= (595.5)^2 = 354620.25
s^2= (1/n-1)[EX^2-(((EX)^2)/n)]
(1/274)[83416.25-((565.5^2)/275)]
(1/274)[83416.25-((354620.25)/275)]
(1/274)[83416.25-(1289.528182)]
(1/274)[82126.72182]= 299.7325614
Variance =299.7325614
Standard Deviation=Square Root of 299.7325614
Which equals 17.31278607
Rounded three decimal places= 17.313
However this answer was incorrect.
Please help!
This is to help the spacing
x = Time (minutes) Number of Customers
0 ≤ x < 1………………………………….. 75
1 ≤ x < 2………………………………….. 58
2 ≤ x < 3…………………………………..64
3 ≤ x < 4………………………………….. 40
4 ≤ x < 5………………………………….. 38
I tried
Frequency Midpoint (xm) f*xm (f*xm)^2
75—————-.5———— 37.5——140625
58—————-1.5———— 87 —— 7569
64—————-2.5———— 160 ——25600
40—————-3.5———— 140——19600
38—————-4.5 ———— 171 ——29241
+------ +------ +-------
275——————————595.5-—83416.25
To find the standard deviation of the amount of time spent in line, you need to follow these steps:
1. Find the midpoint for each class interval. The midpoint is the average of the upper and lower class limits. Here are the values for each category:
First category: (0 + 1) / 2 = 0.5
Second category: (1 + 2) / 2 = 1.5
Third category: (2 + 3) / 2 = 2.5
Fourth category: (3 + 4) / 2 = 3.5
Fifth category: (4 + 5) / 2 = 4.5
2. Multiply each midpoint by its corresponding frequency to obtain f * xm. Here are the values for each category:
First category: 0.5 * 75 = 37.5
Second category: 1.5 * 58 = 87
Third category: 2.5 * 64 = 160
Fourth category: 3.5 * 40 = 140
Fifth category: 4.5 * 38 = 171
3. Calculate the sum of the f * xm values:
Sum = 37.5 + 87 + 160 + 140 + 171 = 595.5
4. Calculate the sum of the squares of the f * xm values:
Sum of (f * xm)^2 = (37.5^2) + (87^2) + (160^2) + (140^2) + (171^2) = 140625 + 7569 + 25600 + 19600 + 29241 = 234635
5. Calculate the variance:
Variance = (1 / (n - 1)) * (Sum of (f * xm)^2 - ((Sum of (f * xm))^2 / n))
Variance = (1 / (275 - 1)) * (234635 - (595.5^2 / 275))
Variance = (1 / 274) * (234635 - 1309.41)
Variance ≈ 859.1879416
6. Calculate the standard deviation:
Standard Deviation = √(Variance)
Standard Deviation ≈ √(859.1879416)
Standard Deviation ≈ 29.297
Rounded to three decimal places, the standard deviation of the amount of time spent in line is 29.297
To calculate the standard deviation of the amount of time spent in line, you need to follow these steps:
Step 1: Calculate the midpoint (xm) for each class interval.
To find the midpoint, you take the average of the upper and lower limits of each class interval. For example, for the first class interval (0 ≤ x < 1), the midpoint would be (0 + 1) / 2 = 0.5.
Step 2: Calculate f * xm for each class.
Multiply the frequency (f) of each class by its corresponding midpoint (xm). For example, for the first class, 75 * 0.5 = 37.5.
Step 3: Calculate (f * xm)^2 for each class.
Square the value obtained in Step 2 for each class. For example, (37.5)^2 = 1406.25.
Step 4: Sum up the values of (f * xm)^2 for all classes.
Add all the squared values obtained in Step 3 to get the sum. For example, 1406.25 + ... + (value for the last class) = the sum of (f * xm)^2.
Step 5: Calculate the mean (average) of the squared values.
Divide the sum obtained in Step 4 by the total number of observations (in this case, the sum of frequencies). Let's call this value EX^2.
Step 6: Calculate the square of the mean.
Square the mean (EX) obtained in Step 5. Let's call this value (EX)^2.
Step 7: Calculate the variance.
Use the formula: variance = (1 / (n - 1)) * (EX^2 - ((EX)^2 / n)), where n is the total number of observations in this case (sum of frequencies).
Step 8: Calculate the standard deviation.
Take the square root of the variance obtained in Step 7. This will give you the standard deviation.
Now, looking back at your calculations:
Step 4 seems to have been done incorrectly. The sum of (f * xm)^2 should be 354620.25 (which you correctly calculated).
Assuming the sum of frequencies (n) is 275 (you correctly calculated this), let's proceed with the remaining steps:
Step 5: EX^2 = 354620.25 / 275 = 1290.5645455 (approximated to a few decimal places).
Step 6: (EX)^2 = (595.5 / 275)^2 = 1.294089862 (approximated to a few decimal places).
Step 7: variance = (1 / (275 - 1)) * (1290.5645455 - (1.294089862 / 275)) = 4.156190321 (approximated to a few decimal places).
Step 8: standard deviation = square root of variance = sqrt(4.156190321) = 2.038848088 (approximated to a few decimal places).
Rounded to three decimal places, the correct answer would be 2.039.