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
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.)
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