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Properties of the sampling distribution of x̄: mean μ, standard error σ/√n, and shape via the Central Limit Theorem.
Learn step-by-step with practice exercises built right in.
The is the distribution of all possible values of when repeatedly drawing samples of size from a population. Understanding this distribution is the foundation of inference.
For a population with mean and standard deviation , when drawing samples of size :
Mean of the sampling distribution: The sample mean is an unbiased estimator of the population mean.
Standard error (standard deviation of ): Larger samples produce less variability in .
Shape of the sampling distribution:
The Central Limit Theorem states: No matter the shape of the population distribution, the sampling distribution of approaches a Normal distribution as increases.
Practical implications:
The standard error decreases with the square root of :
This means:
A population has and (Normal). Draw samples of size . What is the sampling distribution of ?
Solution:
So .
Find :
About 15.87% of sample means exceed 103.
A population is heavily skewed with and . Draw samples of size . Describe the sampling distribution of .
Solution:
So approximately.
Find :
About 38.3% of sample means fall between 49 and 51.
| Condition | Requirement |
|---|---|
| Random sampling | Sample must be randomly selected from the population. |
| Independence | When sampling without replacement, (10% condition). |
| Normality / CLT | Population is Normal, OR (or sample data shows approximate normality). |
⚠️ Confusing and : The population SD is ; the standard error is . They are very different! Standard error decreases as increases; population SD does not.
⚠️ Forgetting the Square Root of : When computing standard error, always divide by , not by .
⚠️ CLT Threshold: While is a rule of thumb, it's not a hard cutoff. If the population is already Normal, CLT applies for any . If the population is very skewed, you may need .
💡 TI-84 / TI-Nspire: Use normalcdf() to find probabilities for . For with and , enter: normalcdf(-999999, 103, 100, 3).
A population has and . Samples of size are drawn. Find and .
Mean of sampling distribution:
Standard error:
A population is Normally distributed with and . A sample of is drawn. Find .
A skewed population has and . Samples of are drawn. By the Central Limit Theorem, approximately what percentage of sample means fall within units of the population mean?
Avoid these 3 frequent errors
Answer: and .
Sampling distribution:
Standardize:
Find probability:
About 4.75% of sample means are below 190.
By CLT (since ):
Find :
Probability:
About 95.45% of sample means fall within units (or within standard errors).