Math Problem Statement
Consider a sample with six observations of 16, 11, 13, 22, 15, and 19. 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.)
Solution
To compute the z-score for each observation in the sample, we use the formula:
Where:
- is the observation,
- is the mean of the sample,
- is the standard deviation of the sample.
Step 1: Compute the mean () of the sample
The sample is .
Step 2: Compute the standard deviation ()
The standard deviation is computed using the formula for the sample standard deviation:
Where:
- is each observation,
- is the mean,
- is the number of observations.
Let's calculate it:
Summing these squared differences:
Now, compute the standard deviation:
Step 3: Compute the z-scores
Now that we have and , we can compute the z-score for each observation:
-
For :
-
For :
-
For :
-
For :
-
For :
-
For :
Final z-scores:
- For , z-score = 0
- For , z-score = -1.25
- For , z-score = -0.75
- For , z-score = 1.50
- For , z-score = -0.25
- For , z-score = 0.75
Would you like any more details or have any further questions?
Here are 5 related questions to deepen your understanding:
- What is the significance of a z-score in statistics?
- How does changing the sample size affect the standard deviation?
- Can z-scores be used to compare values from different datasets?
- What are the properties of a normal distribution in terms of z-scores?
- How does skewness in a dataset affect the z-scores?
Tip: A z-score helps determine how far a data point is from the mean in units of standard deviation.
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Math Problem Analysis
Mathematical Concepts
Statistics
Descriptive Statistics
Z-score
Standard Deviation
Mean
Formulas
z = (x - μ) / σ
Mean (μ) = Σx / n
Standard Deviation (σ) = sqrt(Σ(x_i - μ)^2 / (n - 1))
Theorems
Empirical Rule
68-95-99.7 Rule
Suitable Grade Level
Grades 9-12
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