Real-World Applications
A survey reports a sample mean of 52 with a standard error of 3. This is what the standard error tells us.
What is the typical distance the sample mean is from the true population mean?
This is the distribution of a statistic, such as the mean or proportion, over many samples.
What is a sampling distribution?
This theorem says the distribution of sample means approaches normal as sample size increases.
What is the Central Limit Theorem?
This is the name of the standard deviation of a sampling distribution.
What is the standard error?
Increasing the sample size reduces this measure of spread.
What is the standard error?
A factory samples 40 batteries daily. This allows use of the CLT to justify a normal model because of this reason.
What is a sufficiently large sample size?
This statistic is the average of values from a sample.
What is the sample mean?
According to the CLT, the sampling distribution will be approximately this shape.
What is normal?
This formula computes SE of the sample mean when σ is known: ____.
What is σ / √n?
As sample size increases, this happens to the standard error.
What is it decreases?
This quantity, commonly seen in confidence intervals, is directly influenced by the standard error.
What is the margin of error?
This statistic estimates the proportion of successes in a sample.
What is the sample proportion?
This is the usual minimum sample size needed for the CLT to apply to a skewed distribution.
What is 30?
This formula gives the SE of a sample proportion.
What is √[p(1-p)/n]?
A larger sample size causes the sampling distribution to become more like this shape.
What is normal?
A student says, "The sampling distribution gets narrower because the population is smaller." What’s wrong with that?
What matters is sample size, not population size (as long as the population is large enough or ≥10× sample size).
This term describes a number that summarizes a population.
What is a parameter?
The CLT also applies to this type of statistic when np and n(1−p) ≥ 10.
What is the sample proportion?
This is the main reason standard error is used in confidence intervals and significance tests.
What is to measure sampling variability?
A sample is considered "large enough" for sample proportions if both np and n(1−p) exceed this value.
What is 10?
A random sample of 100 students had an average GPA of 3.4. The population SD is 0.6. What is the probability the sample mean GPA is greater than 3.5?
What is approximately 0.0475? (Standard error = 0.6/√100 = 0.06; z = (3.5−3.4)/0.06 = 1.67 → P ≈ 0.0475)
A statistic that consistently estimates the population parameter is said to have this property.
What is unbiasedness?
The CLT doesn’t apply well when the population has this extreme characteristic and the sample size is small.
What is strong skew or heavy tails?
A smaller SE implies this about the sample mean’s precision.
What is greater precision or less variability?
Even with large n, sampling distributions may not be normal if the population is extremely this.
What is skewed or non-normal?