Consider a sample with six observations of 13, 15, 12, 9, 11, and 12. Compute the z-score for each observation. (Leave no cells blank - be certain to enter "0" wherever required. Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)
To compute the z-score for each observation, we need to use the formula:
z = (x - μ) / σ
where x is the observation, μ is the mean of the sample, and σ is the standard deviation of the sample.
First, let's calculate the mean of the sample:
mean = (13 + 15 + 12 + 9 + 11 + 12) / 6 = 72 / 6 = 12
Next, we'll calculate the standard deviation of the sample:
Step 1: Calculate the deviations from the mean for each observation:
deviation1 = 13 - 12 = 1
deviation2 = 15 - 12 = 3
deviation3 = 12 - 12 = 0
deviation4 = 9 - 12 = -3
deviation5 = 11 - 12 = -1
deviation6 = 12 - 12 = 0
Step 2: Calculate the squared deviations:
squared_deviation1 = (1)^2 = 1
squared_deviation2 = (3)^2 = 9
squared_deviation3 = (0)^2 = 0
squared_deviation4 = (-3)^2 = 9
squared_deviation5 = (-1)^2 = 1
squared_deviation6 = (0)^2 = 0
Step 3: Calculate the sum of squared deviations:
sum_of_squared_deviations = 1 + 9 + 0 + 9 + 1 + 0 = 20
Step 4: Calculate the variance:
variance = sum_of_squared_deviations / (sample size - 1) = 20 / (6 - 1) = 20 / 5 = 4
Step 5: Calculate the standard deviation:
standard_deviation = √variance = √4 = 2
Now that we have the mean (12) and the standard deviation (2), we can calculate the z-score for each observation:
z1 = (13 - 12) / 2 = 1 / 2 = 0.50
z2 = (15 - 12) / 2 = 3 / 2 = 1.50
z3 = (12 - 12) / 2 = 0 / 2 = 0
z4 = (9 - 12) / 2 = -3 / 2 = -1.50
z5 = (11 - 12) / 2 = -1 / 2 = -0.50
z6 = (12 - 12) / 2 = 0 / 2 = 0
Therefore, the z-scores for the six observations are as follows:
z1 = 0.50
z2 = 1.50
z3 = 0
z4 = -1.50
z5 = -0.50
z6 = 0
To compute the z-score for each observation, we need to calculate the mean and standard deviation of the sample.
First, let's calculate the mean:
Mean = (13 + 15 + 12 + 9 + 11 + 12) / 6
= 72 / 6
= 12
Next, let's calculate the standard deviation:
Step 1: Calculate the differences between each observation and the mean:
(13 - 12), (15 - 12), (12 - 12), (9 - 12), (11 - 12), (12 - 12)
= 1, 3, 0, -3, -1, 0
Step 2: Square each difference:
1^2, 3^2, 0^2, (-3)^2, (-1)^2, 0^2
= 1, 9, 0, 9, 1, 0
Step 3: Calculate the sum of the squared differences:
1 + 9 + 0 + 9 + 1 + 0
= 20
Step 4: Divide the sum by (n-1), where n is the number of observations:
20 / (6-1)
= 20 / 5
= 4
Step 5: Take the square root of the result:
√4
= 2
Now that we have the mean (12) and standard deviation (2), we can calculate the z-score for each observation:
Z-score = (Observation - Mean) / Standard Deviation
Z-score for 13 = (13 - 12) / 2
= 1 / 2
= 0.50
Z-score for 15 = (15 - 12) / 2
= 3 / 2
= 1.50
Z-score for 12 = (12 - 12) / 2
= 0 / 2
= 0
Z-score for 9 = (9 - 12) / 2
= -3 / 2
= -1.50
Z-score for 11 = (11 - 12) / 2
= -1 / 2
= -0.50
Z-score for 12 = (12 - 12) / 2
= 0 / 2
= 0
Therefore, the z-scores for each observation are: 0.50, 1.50, 0, -1.50, -0.50, 0.