Chpt 7 - Sampling Distributions Flashcards
The population mean (μ) is a
Parameter/statistic
and is
not fixed/fixed
The population mean (μ) is a parameter and is a fixed number
The sample mean (x̄) is a
Parameter/statistic
and is
not fixed/fixed
The sample mean (x̄) is a statistic and is not fixed
Why is the sample mean (x̄) not a fixed number?
Because it’s values are means of the samples randomly selected from the population, so it changes form sample to sample
Why is the sample mean a continuous random variable?
Because its value is NOT fixed and changes depending on the sample that is randomly selected.
What type of variable is a sample mean?
Continuous random variable, because its value is not fixed and depends on the sample that is randomly selected
Because the sample mean (X̄) is not a fixed number, what do we need to know to measure the error that one may make when using a sample mean (x̄) to estimate the population mean?
Which values it can take and how likely it takes a specified value
Long story short, we need to describe the distribution of the sample mean (X̄).
What does the distribution of the sample mean (X̄) depend on?
The sample size of the samples selected
The distribution of the variable under consideration
For any sample size, if the distribution of X is normal, what is the distribution of the sample mean (X̄)?
Normal
For a larger sample size, if the distribution of X is not normal, what is the distribution of the sample mean (X̄)?
Approximately normal
For a smaller sample size, if the distribution of X is not normal, what is the distribution of the sample mean (X̄)?
Not normal
What is the distribution of the sample mean (X̄) called?
Sampling distribution of the sample mean
What is the distribution of the random variable under consideration X called?
Parent distribution
What is the central limit theorem (CLT)?
For a relatively large sample size n, the sample mean (X̄) is approximately normally distributed regardless of the parent distribution
How large should n be to be large enough if the parent distributions is not too extremely skewed?
n>30
Assume that the random variable X under consideration has a mean (μ) and standard deviation (σ). What is the mean of X̄ of samples with sample size n? What about the standard deviation?
μx̄ = μ
σx̄ = σ/√n
If the parent distribution is normal with mean (μ) and standard deviation (σ), what is the shape, mean and standard deviation of the sampling distribution?
Normal shape
μx̄ = μ
σx̄ = σ/√n
If the parent distribution is not normal with mean (μ) and standard deviation (σ), what is the shape, mean and standard deviation of the sampling distribution when the sample size is large?
Shape is approximately normal
μx̄ = μ
σx̄ = σ/√n
If the parent distribution is not normal with mean (μ) and standard deviation (σ), what is the shape, mean and standard deviation of the sampling distribution when the sample size is small?
Shape cannot be determined
μx̄ = μ
σx̄ = σ/√n
The standard deviation of X̄ is determined by the sample size n. The larger the sample size n, the ______ the standard deviation of X̄.
smaller
A variable of a population has a mean of 40 and standard deviation of 2.
If the variable is normally distributed, what is the sampling distribution of the sample mean for samples of size 4?
Normal distribution
μx̄ = μ = 40
σx̄ = σ/√n = 2/√4 = 2/2 = 1
A variable of a population has a mean of 40 and standard deviation of 2.
If the variables distribution is unknown, what is the sampling distribution of the sample mean for samples of size 4?
Distribution cannot be determined (sample size is too small)
μx̄ = μ = 40
σx̄ = σ/√n = 2/√4 = 2/2 = 1
A variable of a population has a mean of 40 and standard deviation of 2.
If the variables distribution is unknown, what is the sampling distribution of the sample mean for samples of size 100?
Approximately normal
μx̄ = μ = 40
σx̄ = σ/√n = 2/√100 = 2/10 = 0.2
What is the rule of thumb for deciding whether the sample size is relatively large such that the sample mean is approximately normally distributed?
n > 30
Roughly speaking, what property of the distribution of the variable under consideration determines how large the sample size must be for a normal distribution to provide an adequate approximation to the distribution of the sample mean?
Skewness
a larger skew needs a larger sample size
What is the chance that the sample mean and population mean are exactly the same?
0, a sampling error always exists
What is a sampling error?
The difference between the sample mean and population mean
Suppose the population mean of the parent distribution is unknown. A sample mean of a sample with n=2 and a sample mean of a sample with n=30 are obtained to estimate the population mean. Which one is more likely to be closer to the population mean, i.e., have smaller sampling error when used for estimating the population mean?
The larger sample size, so n=30
How do we determine the probability that the sampling error made in estimating the population mean will be within a certain range, such as +/-1
- Determine the sample distribution shape, mean, and standard deviation
- Standardize the x values for the +/- range to find the z values
- Use table 2 to convert the z values to the probability
- Subtract - value from the + value to determine the probability that the sample mean falls within the +/- value
Say we are attempting to determine the sampling error is within +/-1. How would you standardize the values for x?
Z+ = (X+1-μ)/σ
Z- = (X-1-μ)/σ
Determine the probabilities for both z values
P(Z+) - P(Z-) = P(μ of sample +/-1)
