stats exam 2
alpha level critical region
alpha level- level of significance critical region- contains extreme sample means that are very unlikely if null is true
Why is the mean of a distribution of z-scores always equal to 0?
because the definition of z-scores says that all positive scores are above the mean and all negative scores are below the mean (in other words, for z-scores, 𝞵= 0)
Which of the following is an accurate definition for the power of a statistical test? a) the probability of rejecting a true null hypothesis b) the probability of supporting a false null hypothesis c) the probability of rejecting a false null hypothesis d) the probability of supporting a true null hypothesis
c) the probability of rejecting a false null hypothesis
purpose of power:
calculated before study to determine whether it is worth time/ money/ effort
as sample size increases the standard error ________________ the standard error can never be larger than the population standard deviation if the sample size is the smallest sample size possible (n=1) the standard error and population standard deviation are ____________
decreases; the same
What is a hypothesis test, and what is the purpose of conducting one?
inferential statistical procedure that uses sample stats to evaluate the validity of a hypothesis about a population parameter
What is probability?
likelihood of a specific outcome, defined as a proportion/fraction of all possible outcomes
effect size (d)
measures the absolute magnitude of treatment effect, regardless of sample size
STANDARD ERROR FORMULA:
oM = o/sq.root:n
How do you calculate the probability of a certain outcome? What information do you need in order to do this calculation?
p= (# of outcomes classified)/ (total number of possible outcomes)
increased sample size = ___________ standard error
reduced
Be able to interpret a z-score (both the sign and the number).
sign if z>0 : score is greater than mean if z<0 : score is less than mean if z=0 : score is equal to mean number if z=1 : score is 1 standard deviation (SD) above mean if z=-1 : score is 1 SD below mean
As sample size increases, the expected value of M ________.
stays constant
When the sample size is greater than n = 30 ________.
the distribution between sample MEANS will be approximately normal
power
the probability that the hypothesis test will correctly reject a false null hypothesis
What is the standard error?
variability in a distribution of sample means -It provides a measure of how much difference is expected from one sample to another. If it's small, than the means are close together and have similar values. -If it's large, the sample means are scattered over a wide range with big differences from one sample to another.
What does it mean to standardize a distribution? Why would you want to do this?
-transforming any distribution (with any mean and standard deviation) into a distribution with a mean of 0 and a standard deviation of 1 -standardized distribution: composed of scores that have been transformed to create predetermined values for the mean and deviation -why? these types of distributions are used to make dissimilar distributions comparable
Power =
1 - ß ß: probability of a type II error
Distribution of sample means is almost perfectly normal is EITHER of the following is true: 1. Population from which samples are selected is a ____________________ 2. Number of scores (n) in each sample is ______+
1. Population from which samples are selected is a normal distribution 2. Number of scores (n) in each sample is 30+
How do you construct a distribution of sample means? When you do so, what characteristics should you expect to see in the distribution?
1. Select a random sample of a specific size (n) from a population 2. calculate the sample mean, place the sample mean in a frequency distribution. 3. Select another random sample with the same number of scores. 4. Again, calculate the sample mean and add it to your distribution **sample means should pile up around population mean **pile of sample means should be in normal distribution
What are the two requirements for random sampling?
1. Simple random sample: each individual in the population has equal chance of being selected 2. Independent random sampling: if more than one individual is selected, probability must stay constant from one selection to the next
Steps to conducting a hypothesis test:
1. State hypothesis (alternative hypothesis, null hypothesis) 2. Set the criteria for a decision (alpha level, critical region) 3. collect data & compute sample statistics 4. make a decision! (reject null = statistically significant)
2 basic purposes of standard error
1. describes how similar/different sample means are -Small standard error --> sample means are close together -Large standard error --> sample means are far apart 2. describes how well a sample mean represents the entire distribution
magnitude of standard error depends on 2 factors: 1. sample size 2. population standard deviation size
1. sample size -law of large numbers: the larger the sample size (n) is, the more likely it is that the sample mean (M) is close to the population mean (u) 2. population standard deviation size -Smallest possible sample (and largest standard error) occurs when n = 1 -Each sample = single score -Distribution of sample means = distribution of individual scores -When n=1, oM= o = (standard error = population standard deviation) o Standard deviation: starting point for standard error o When n > 1 sample becomes better representation of population standard error decreases
What are the two steps involved with standardizing a distribution with new values for μ and σ?
1. transform raw scores (X) to z-scores 2. transform z-scores into new raw scores so that specific u and o are attained
What is the unit normal table? What information is available on the unit normal table.
A table listing proportions corresponding to each z-score location in a normal distribution. Z scores and proportions.
What is the central limit theorem? What does the central limit theorem tell you about a distribution of sample means? (Make sure to address shape, central tendency, and variability in your answer.)
Central Limit Theorem: gives description of sample means for ANY given population For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of o/√n. -Describes the distribution of sample means for any population, no matter what shape, mean, or standard deviation -The distribution of sample means "approaches" a normal distribution very rapidly. By the time the sample size reaches n=30, the distribution is almost perfect normal. -Shape: Perfectly normal if: the population from which the samples are selected is a normal distribution or the number of scores (n) in each sample is relatively large, around 30 or more.
What is random sampling, and why is it important?
Drawing the sample so that each individual has an equal chance of being selected and the probability stay the same for each person. This eliminates bias when selecting the sample.
Small alpha level ➞ increased risk of type ___ errors Large alpha level ➞ increased risk of type __ errors
II I
Factors that increase power: Factors that decrease power:
INCREASE POWER: -increasing sample size -changing from two-tailed test to one-tailed test DECREASE POWER: -decreasing alpha level
Looking ahead to inferential statistics: How are z-scores helpful when conducting inferential statistics?
Inferential statistics: using samples to make statements about populations Extreme z-scores ➞ conclude that treatment had an effect
What is the distribution of sample means? What is the research situation in which the distribution of sample means is required?
PURPOSE: find probability of any given sample mean The distribution of sample means is the collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from a population. It allows us to predict the characteristics of a sample with some accuracy. Once we have specified the complete set of all possible sample means (i.e. the distribution of sample means) we can find the probability of selecting any specific sample mean.
What is a raw score? What is a z-score? What are the advantages on transforming a raw score into a z-score?
Raw Score - original, unchanged score that is direct result of measurement (X) Z- score - number of standard deviations that a raw score is above/below the distribution mean 2 main purposes of z-scores: transforming raw scores into scores that have more information specifically, the exact location of the X value in the distribution standardize distributions (allow comparison to other distributions that have also been transformed into z-scores)
What is sampling with replacement, and how does it relate to the requirements for random sampling?
Returning selected individual to the population before the next selection This keeps the probability the same on each selection
What potential problems arise when studying samples? (Make sure to include the term "sampling error".)
Sampling error is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. If you take two samples from the same population, the samples are different.
What is the difference between standard error and standard deviation?
Standard Error- distance between SAMPLE MEAN and population mean (M-u) Standard deviation- distance between SCORE and population mean (x-u)
What is the difference between standard error and sampling error?
Standard Error: The standard distance between a sample mean and the population mean. (in other words the standard deviation of the distribution of sample means) Sampling Error: is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter (in other words the discrepancy between a statistic and a parameter)
Describe the four steps of conducting a hypothesis test.
Step 1 questions: What is the difference between the null hypothesis and the alternative hypothesis? Step 2 questions: What is the alpha level/level of significance? What is the critical region? How are alpha level and critical region used to set the criteria for a decision? Step 3 questions: How are z-scores used in a hypothesis test? Step 4 questions: What does it mean to reject the null hypothesis? What does it mean to fail to reject the null hypothesis? Why do we approach this issue in terms of rejecting/failing to reject the null hypothesis?
How do you know what shape a distribution of z-scores will take?
The distribution of z-scores will have exactly the same shape as the original distribution of scores (ex: if original distribution is negatively skewed then the z-score distribution will also be negatively skewed)
How does probability relate to frequency distributions? Specifically, how can you determine the probability of randomly selecting a particular score from a frequency distribution?
The frequency distribution graph is able to represent probabilities as proportions of the graph.