Susan keeps track of the number of tickets sold for each play presented at the community theater. Within how many standard deviations of th mean do all the values fall? 135, 71, 69, 80, 158, 152, 161, 96, 122, 118, 87, 85.
3
4
5
2
I keep coming up with 1 as the answer and that's not a choice?
To find how many standard deviations from the mean all the values fall, we need to calculate the standard deviation of the given data set.
First, we need to find the mean (average) of the data set:
Mean = (135 + 71 + 69 + 80 + 158 + 152 + 161 + 96 + 122 + 118 + 87 + 85) / 12
Mean = 1244 / 12
Mean = 103.67 (approximated to two decimal places)
Now, let's calculate the standard deviation:
1. Calculate the deviation from the mean for each value:
Deviation from the mean = Value - Mean
Deviation from the mean for each value:
135 - 103.67 = 31.33
71 - 103.67 = -32.67
69 - 103.67 = -34.67
80 - 103.67 = -23.67
158 - 103.67 = 54.33
152 - 103.67 = 48.33
161 - 103.67 = 57.33
96 - 103.67 = -7.67
122 - 103.67 = 18.33
118 - 103.67 = 14.33
87 - 103.67 = -16.67
85 - 103.67 = -18.67
2. Square each deviation:
(31.33)^2 ≈ 982.25
(-32.67)^2 ≈ 1065.29
(-34.67)^2 ≈ 1200.81
(-23.67)^2 ≈ 558.49
(54.33)^2 ≈ 2951.09
(48.33)^2 ≈ 2332.89
(57.33)^2 ≈ 3289.29
(-7.67)^2 ≈ 58.93
(18.33)^2 ≈ 336.09
(14.33)^2 ≈ 205.29
(-16.67)^2 ≈ 277.89
(-18.67)^2 ≈ 348.49
3. Find the average of the squared deviations:
Average of squared deviations = (982.25 + 1065.29 + 1200.81 + 558.49 + 2951.09 + 2332.89 + 3289.29 + 58.93 + 336.09 + 205.29 + 277.89 + 348.49) / 12
Average of squared deviations = 15057.42 / 12
Average of squared deviations ≈ 1254.78 (approximated to two decimal places)
4. Take the square root of the average of squared deviations:
Standard deviation = √(1254.78)
Standard deviation ≈ 35.43 (approximated to two decimal places)
Now, we need to determine within how many standard deviations from the mean all the values fall. To do this, we calculate the z-score for each value.
Z-score = (Value - Mean) / Standard Deviation
Z-score for each value:
31.33 / 35.43 ≈ 0.88
-32.67 / 35.43 ≈ -0.92
-34.67 / 35.43 ≈ -0.98
-23.67 / 35.43 ≈ -0.67
54.33 / 35.43 ≈ 1.53
48.33 / 35.43 ≈ 1.37
57.33 / 35.43 ≈ 1.62
-7.67 / 35.43 ≈ -0.22
18.33 / 35.43 ≈ 0.52
14.33 / 35.43 ≈ 0.40
-16.67 / 35.43 ≈ -0.47
-18.67 / 35.43 ≈ -0.53
Now we need to determine the range that includes all the z-scores calculated.
Looking at the z-scores, we can see that the largest absolute value is 1.62. This means that all the values fall within 1.62 standard deviations of the mean.
Therefore, the answer is 2.
To find the number of standard deviations within which all the values fall, we first need to calculate the mean and standard deviation of the given set of values.
Step 1: Calculate the mean
To find the mean (average) of a set of values, add up all the values and divide the sum by the total number of values.
Mean = (135 + 71 + 69 + 80 + 158 + 152 + 161 + 96 + 122 + 118 + 87 + 85) / 12
Mean = 1356 / 12
Mean = 113
Step 2: Calculate the standard deviation
To calculate the standard deviation, we need to follow these steps:
1. Find the difference between each value and the mean.
2. Square each difference.
3. Calculate the mean of the squared differences.
4. Take the square root of the mean of the squared differences.
Let's calculate the standard deviation step by step:
Difference from mean for each value:
(135 - 113) = 22
(71 - 113) = -42
(69 - 113) = -44
(80 - 113) = -33
(158 - 113) = 45
(152 - 113) = 39
(161 - 113) = 48
(96 - 113) = -17
(122 - 113) = 9
(118 - 113) = 5
(87 - 113) = -26
(85 - 113) = -28
Squared differences for each value:
22^2 = 484
(-42)^2 = 1764
(-44)^2 = 1936
(-33)^2 = 1089
45^2 = 2025
39^2 = 1521
48^2 = 2304
(-17)^2 = 289
9^2 = 81
5^2 = 25
(-26)^2 = 676
(-28)^2 = 784
Calculate the mean of the squared differences:
Mean of squared differences = (484 + 1764 + 1936 + 1089 + 2025 + 1521 + 2304 + 289 + 81 + 25 + 676 + 784) / 12
Mean of squared differences = 11088 / 12
Mean of squared differences = 924
Take the square root of the mean of squared differences:
Standard deviation = √924
Standard deviation ≈ 30.39
Since all the values fall within 3 standard deviations of the mean, the answer is 3.
Therefore, the correct answer is 3.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.