chapter 9 psych stats
Center:
At least in our simulations, the mean of the sampling distribution of means seems to have the same mean as the hypothesized population distribution.
shape of sampling distribution
Regardless of the hypothesized shape of the population distribution (e.g., die throws were uniform in shape, thumb lengths were normal) the sampling distribution tends to have a normal shape.
How can sampling distributions help us interpret our data?
Sampling distributions give us a way to assess whether a relationship we've observed in our data is likely to have occurred just by chance.
Spread:
Sampling distributions of the mean appear to be less variable (less spread out) than the population distributions that they come from.
How can we figure out what the expected population mean should be of rolling a die
Simulate a very large sample of die rolls and calculate the mean. The mean is the average of these equally likely outcomes: 1, 2, 3, 4, 5, 6.
If the DGP had a larger spread, how would that affect your confidence in your estimate?
That would lead to less confidence in my estimate.
Which of the following are definitions of "standard error?" (Check all that apply.) The average distance from the estimate to the parameter The standard deviation of a sample The standard deviation of a sample mean The standard deviation of many possible means The standard deviation of a sampling distribution The standard deviation of the population distribution
The average distance from the estimate to the parameter The standard deviation of many possible means The standard deviation of a sampling distribution
Which of the following might be a good measure of how close all the estimates are to the parameter?
The average distance from the estimates to the parameter
Which part of the Central Limit Theorem corresponds to this idea that estimates cluster around the parameter?
The mean of the population will also be the mean of the distribution of means.
If we take random samples of n=1 from the population and calculate the mean of each sample (which is just the number we sampled divided by 1), we end up with a sampling distribution that looks exactly like the population. If the population's standard deviation is represented by σ, then what is your prediction for the standard error of the sampling distribution of the mean when n=1? The same value as σ 1 s The only way to tell is to conduct a simulation 𝜎√n 𝜎√1
The same value as 𝜎 𝜎𝑛√σn 𝜎1√
What is standard error
The standard deviation of the sampling distribution just from this example of the sampling distribution of variances,
Which part of the Central Limit Theorem corresponds to this idea that estimates are on average closer to the parameter when sample size is larger?
The standard error will be inversely proportional to the square root of the sample size.
Imagine we drew two random samples from a population, and measured each case sampled on the same outcome variable. One sample had an n = 30, the other an n = 60. Which of the following statements is true?
The sum of squares of the larger sample would almost certainly be greater than the sum of squares of the smaller sample.
What would happen if the DGP produces more variation?
The variation in the sampling distribution would be larger. The variation in the population would be larger. The variation in the sample would be larger.
What have we learned about the shape of the sampling distribution of means from this video?
The wildly different parent populations (e.g., uniform, skewed, crazy) from the video produced sampling distributions of the mean that all had a roughly normal shape.
What did we find out about the standard error part of the Central Limit Theorem (CLT)
This is the mathematical relationship between the standard error, population standard deviation, and sample size. By using the CLT, you can predict the standard error you might see from simulations pretty well.
How likely is it that you would generate a sample mean of 6? That is, how likely is it that you would roll 24 6s in a row?
Unlikely, even though it is possible.
Which of these questions can be approximately answered by examining this simulated population?
What is the likelihood of selecting an individual with a thumb length greater than 70 mm?
you're interested in females' ratings of males' intelligence. You simulate 500 samples of 276 ratings, calculate the mean of each sample, and plot the resulting distribution of means in a histogram. What will be the mean of this sampling distribution?
Whatever mean you set when you ran the simulation
Which of the following best describes how you would read this conditional probability? P(Earning an A| Complete all assignments)
what is the probability of earning an A given that you've completed all assignments,
Which of these is the mean of the sampling distribution of standard deviations?
𝜇𝑠
Which of these is the standard deviation of the sampling distribution of standard deviations?
𝜎𝑠
If event A and event B are independent events, then which of the following is true? P(A) = P(A | B) P(A) ≠ P(A | B)
P(A) = P(A | B)
which would have the same exact value?
Population mean and sampling distribution mean
Why did we simulate sampling distributions from multiple hypothesized DGPs with different means?
Because the sampling distributions allow us to evaluate how likely our original sample's parameter estimate (60.1 mm) was to have come from the given DGP
sampling distribution
the distribution of an estimate across many possible samples.
What is a sampling distribution made of?
Estimates based on multiple possible samples that could have been generated by a possible DGP.
What have we learned about estimates in general?
Estimates vary less for larger sample sizes.
The Central Limit Theorem
the distribution of sample averages tends to be normal regardless of the shape of the process distribution The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution.
True or False: The probability of getting a job if you graduate from college is the same as the probability of graduating from college if you get a job.
False
What would the sampling distribution of means look like for samples of n=1?
It would have the same shape and standard deviation as the population distribution.
If, in the population, females' mean intelligence ratings of males is 7.7, with a standard deviation of 1.2, how likely is it that we would randomly draw a sample of n=276 with a mean of less than 7.5?
Less than 5%
Which two parameters define a particular normal distribution?
Mean and Standard deviation
Even if the population has a particular mean and standard deviation, will every other sample also have the same mean and standard deviation? Why or why not?
No, the samples will vary because of sampling variation.
what have we learned about distributions of other estimates besides the mean?
Only sampling distributions of the mean tend to be normal with larger sample sizes. Distributions of other estimates don't reliably follow this pattern.
Central Limit Theorem.
describes the shape, center, and spread of a distribution of sample means of equal size when each sample is randomly chosen from some population.
Which two of these three lines of R code do the same thing? response - incorrect do(20) * rnorm(1,Thumb.stats$mean, Thumb.stats$sd) rnorm(20,Thumb.stats$mean, Thumb.stats$sd) do(20) * mean(rnorm(157,Thumb.stats$mean, Thumb.stats$sd))
do(20) * rnorm(1,Thumb.stats$mean, Thumb.stats$sd) rnorm(20,Thumb.stats$mean, Thumb.stats$sd)
Let's record our initial predictions. If we draw random samples from a skewed distribution and then build up a sampling distribution of mean, what will be the shape of that sampling distribution?
normal
