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Psyc 300A > Midterm #4 > Flashcards

Midterm #4 Flashcards

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1
Q

What is the standard normal distribution a distribution of?

A

Our correct sampling (comparison) distribution is a distribution of individual scores
-There is a match between the type of sample score we have (X) and the distribution we compare it to (a distribution of individual scores)

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2
Q

What are 3 limitations of the standard normal distribution?

A
  • Only look at score for a single outcome
  • No baseline
  • Could be just about the one participant, not about the experimental manipulation
  • Difference could be due to individual differences in the participant relative to the comparison group
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3
Q

In a single sample design what is our statistic for analysis?

A

X-bar

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4
Q

What is our correct sampling distribution (comparison) for a sample of 50 scores?

A

Distribution of the mean.
There is a match between th type of sample score we have (X-bar) and the distribution we compare it to (sampling distribution)

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5
Q

What is a sampling distribution of the mean vs a sampling distribution?

A

Sampling distribution of the mean= the distribution of sample means over repeated sampling from one population.

Sampling distribution= a distribution of a statistic (mean, SD, etc.) over repeated sampling.

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6
Q

Suppose there is a jar containing many gumballs,
each with a unique number on it. The numbers range
from 0 to 32 and there is an equal number of gumballs
with each number.
A student set out running an experiment with the
following procedure: Pick five gumballs from the jar,
calculate the mean of the numbers on the gumballs,
write down the result on a piece of paper, and put the
gumballs back to the jar. Repeat the process 499
times so altogether there are 500 means recorded.
Then draw a frequency distribution of the 500 means.
In this case, the sample size is
a) 500
b) 499
c) 32
d) 5

A

d)5

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7
Q

Suppose there is a jar containing many gumballs,
each with a unique number on it. The numbers range
from 0 to 32 and there is an equal number of gumballs
with each number.
A student set out running an experiment with the
following procedure: Pick five gumballs from the jar,
calculate the mean of the numbers on the gumballs,
write down the result on a piece of paper, and put the
gumballs back to the jar. Repeat the process 499
times so altogether there are 500 means recorded.
Then draw a frequency distribution of the 500 means.
In this case, the number of samples is
a) 500
b) 499
c) 32
d) 5

A

a) 500

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8
Q

3 Pool balls- labelled 1 2 or 3.

Compute the probability for
each outcome
Probability of selecting a 1 and then another 1
a) Multiplication Rule
b) Addition Rule
c) Combination Rule
A

a) Multiplication Rule

(1/3)x(1/3)=.1111

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9
Q

Pool Ball example:
Compute the probability for
all related outcomes and make a
probability table
Probability of selecting a 1 and then another 1 OR the
probability of selecting a 1 and then a 2 OR the
probability of selecting a 1 and then a 3 etc….
a) Multiplication Rule
b) Addition Rule
c) Combination Rule

A

c) Combination rule.

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10
Q

What are the 3 characteristics we need to look at once we have constructed a sampling distribution?

A

Mean, spread, shape

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11
Q

What is the mean, variance and standard deviation for the sampling distribution of the mean?

A

Mean: µ X-bar
Variance: σ^2 X-bar
Standard deviation:σ X-bar

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12
Q

What is the shape of the sampling distribution?

A

Normal Distribution

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13
Q

Each sampling distribution will be uniquely named based onL (2)

A
  1. The statistic being analyzed
  2. Size of the sample (sample N)

Eg: Sampling distribution of the MEAN for N=2.

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14
Q

In a single sample design with N=30, what is the
appropriate sampling distribution?
a) Sampling distribution of the mean
b) Sampling distribution of the mean for N=30
c) Standard Normal Distribution
d) Sampling distribution for N=30

A

b) Sampling distribution of the mean for N=30

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15
Q

What is the only info we need to know to determine the characteristics of a sampling distribution of the mean?

A
  1. The Characteristics of the distribution of the population of individuals
  2. The number of scores in each sample
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16
Q

What is the central limit theorum?

A

Factual statement about the distributions of means.

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17
Q

Why does the mean of the population always equal the mean of the sampling distribution of the mean?

A

Because all combinations of means are included.

Each sample is based on a group of randomly selected individuals from the population. The mean will sometimes be higher/lower than the mean of the entire population.

Since there are a large number of samples in the sampling distribution, the high and low balance out.

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18
Q

The population has a mean of 14 and a standard
deviation of 3. The sample size of your sampling
distribution is N=10. What is the mean of the sampling
distribution of the mean ( )?
a) 1.4
b) 4.43
c) 14
d) .949

A

c) 14

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19
Q

What is the standard error of the mean?

A

The variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample.

The standard deviation of a distribution of means is the square root of the variance of the distributions of means.

IT IS ALSO KNOWN AS STANDARD ERROR (SE)

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20
Q

What does the standard error of the mean tell you?

A

Tells you how much the various means in the distribution off means tend to deviate from the mean of the population.

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21
Q

Will a distribution of means be less or more spread out than the distributions of the individuals that the samples were taken from?

A

LESS

If you take a sample of 2 scores, it is less likely that both scores will be extreme. For a random sample to have an extreme mean, two extreme scores would have to be both in the same direction. (eg. both very high or very low).

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22
Q

How does sample size and variability interact?

A

The larger the sample size, the smaller the variance of the sampling distribution of the mean.

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23
Q

The population has a mean of 30 and a standard
deviation of 6. The sample size of your sampling
distribution is N=9. What is the variance of the
sampling distribution of the mean?
a) 3
b) 2
c) 10
d) 12
e) 4

A

e) 4

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24
Q

Regardless of the shame of the population, as the sample size (N) increases, what will happen to the shape of the sampling distribution of the mean?

A

It will approach a normal distribution

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25
If a population has a mild skew what does the sample size have to be to be normally distributed? Moderate? Severe?
``` Mild= N =/> 10/12 to be normal Mod= N=/>16-20 Severe= N=/> 30 ```
26
The sampling distribution of the mean, with N=30, of a moderately negative skewed distribution is: a) Positively skewed b) Negatively skewed c) About normal
c) About normal
27
Why is the property of shape important when you have a single participant or single sample design?
The sampling distribution must be kurtotic to carry inferential analyses tests derivied from the random sampling model of hypothesis testing.
28
The entire student body of 225 students took a test. These test scores have a mean of 75, a standard deviation of 10, and are slightly positively skewed. If you randomly chose 25 of these test scores and calculated the mean over and over again, what could be the mean, standard deviation, and skew of this distribution? a) Mean = 75, SD = 10, Skew = 1.2 b) Mean = 75, SD = 0.67, Skew = 0.8 c) Mean = 80, SD = 2, Skew = -1.2 d) Mean = 75, SD = 2, Skew = about 0 e) Mean = 75, SD =0.67, Skew = about 0
d) Mean=75, SD= 2, Skew=~0
29
The population has a mean of 50 and a standard deviation of 6. The sample (N=25) has a mean of 52 and a standard deviation of 5. What is the mean, µ X-bar of the sampling distribution of the mean for N=25?
50
30
What is “standard error of the mean”? a) The standard deviation of the sampling distribution of the mean b) The standard deviation of the standard normal distribution c) The amount the scores in the population vary from the mean d) The difference between the mean of your first sample and the mean of your second sample
a) The standard deviation of the sampling distribution | of the mean
31
What is the mean of a sampling distribution for the central limit theorem?
The mean of a sampling distribution of the mean is the same as the mean of the population of individuals. µ X-bar= µ
32
What is the variability aspect of the central limit theorem?
2a) The variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample. 2b) The standard deviation of a distribution of means is the square root of the variance of the distributions of means. Also known as STANDARD ERROR OF THE MEAN
33
What is the standard error of the mean also known as?
Standard error (SE) Tells you how much the various means in the distribution off means tend to deviate from the mean of the population.
34
Suppose there is a jar containing many gumballs. Each gumball has an integer number from 0 to 32 on it. There is the same number of gumballs with each number on it. As a result, the numbers are distributed uniformly (rectangular distribution) and the mean of all the numbers in the jar is 16. You draw 10 gumballs from the jar and calculate the mean and put the gumballs back in the jar. You then draw 25 gumballs and again calculate the mean. Which mean is more likely to be less than 11? a) The mean of the 10 gumballs is more likely to be < 11 b) The mean of the 25 gumballs is more likely to be < 11 c) Both have roughly the same chance.
a) The mean of the 10 gumballs is more likely to be < 11
35
Three features for variability
1. Sample size affects the spread (variability) of the distribution of sample means. 2. The smaller the sample size, the more spread out of the distribution. 3. Therefore, the mean of a small sample size is more likely to deviate greatly from the mean of 16 (population).
36
How does the sample size effect the variability of the sampling distribution of the mean?
The larger the sample size, the smaller the variability of the sampling distribution of the mean.
37
There are the same number of gumballs with the number 0, the number 1, etc. Since the numbers are evenly distributed in the jar, the distribution is called a uniform (rectangular) distribution. This distribution ranges from 0 to 32, and has a mean of 16. Samples of 10 are taken 10,000 times and the means plotted in a histogram. These 10,000 sample means are called a secondary distribution. The shape of the secondary distribution will be approximately a uniform (rectangular) distribution but may differ by chance variation. True or False?
False
38
How does the shape of the population and the shape of the sampling distribution relate?
Regardless of the shape of the population, as sample size (N) increases, the shape of the sampling distribution of the mean approaches a normal distribution.
39
When do we use a z-test for a single sample design?
When we have a group of participants and when we know or can calculate the population mean (μ) and standard deviation (σ)?
40
When performing a z-test when sigma is known, what is the model hypothesis testing?
Random sampling
41
When performing a z-test when sigma is known, what is the type of data?
Score (interval/ratio)
42
When performing a z-test when sigma is known, what is the statistic for analysis?
X-bar sample mean
43
What is important to ensure when looking at the sample scores and the sampling distribution?
There needs to be a match between the type of sample score we have and the distribution we compare it to (our sampling distribution).
44
What is the sampling distribution for single sample design where sigma is known?
Distribution of means for a given N. (Sampling Distribution for the Mean for N=___)
45
What is the sampling distribution of the mean?
The distribution of sample means over repeated sampling from one population.
46
Given the information below, what is the mean and the standard error of the sampling distribution of the mean for N=25? Population parameters: μ= 500 mm^3 and σ= 100 mm^3
b) µ-X-bar=500 | c) σ X-bar= 20
47
What would z(crit) be if we were to set our critical | value to α = .01 2-tailed?
c) +/- 2.575
48
What does the test ration do?
It takes raw score values and transforms them into a standard score so that test statistic can be compared to a standardized probability distribution.
49
What is p(obs) for z(obs)=2.75, two-tailed?
d) 0.006 Z-table: Look up 2.75, smaller portion. Multiply this value by 2 reflects the 2-tailed hypothesis.
50
What are the three distinct distributions we look at?
Population Distribution Sample Distribution Sampling Distribution
51
When does the shape of the population distribution matter?
May be any shape from rectangular to a normal curve. BUT when N of sample size is small, it is preferable that the shape be at least symmetrical, not skewed. Data represents a collection of all values (X) for members of the population (as identified by the researcher)
52
What is the shape and the statistic of interest for analysis of interest?
Shape: may be any shape: skewed, bell-shaped, flat (rectangular) Statistic of interest for analysis is the sample mean: X-bar
53
What does the sampling distribution of the mean represent? How is the distribution generated?
All possible means for a fixed N (N= size of the sample) and the probability of getting each mean. Based on the size of the sample (N).
54
When is the shape of the sampling distribution of the mean approximately normal?
If samples have 30 or more individuals in the sample of if the population is normal.
55
What is the central tendency of the sampling distribution of the mean?
Mean of the sampling distribution of the mean, µ x-bar
56
What is the variability of the sampling distribution of the mean?
Standard error of the mean, σ x-bar
57
What information do we need for a one-sample z-test?
Need info on both mean and SD of the population.
58
For a one-sample z-test what are the | conditions/assumptions?
1. Need info on both mean and SD of the population. 2. The behaviour we are studying (the DV) is normally distributed in the population. 3. Participants in the sample were randomly sampled from the population.
59
Under what circumstance can we still do a onesample z-test if our DV is not normally distributed in the population? a) If our sampling distribution is normally distributed b) If we have ≥ 30 participants in our sample c) If we know μ and σ d) If the participants were randomly sampled from the population e) A and B
e) A and B
60
What does the t-test allow us to do? (3)
Compare before and after in the same group of participants. Compare an experimental to a control group. Compare 2+ groups of people (eg. different cultures, gender)
61
When is a single sample t-test used?
- When you have scores for a sample of individuals | - When you want to compare this to a population with a known mean (μ) but unknown variance/SD (σ^2/σ)
62
Better-than-average-effect (BTAE)
People evaluate themselves as better than their peers on desirable trains. Eg. 90% of drivers think they are better than the average driver. Most students think they are more intelligent than the average student
63
Do prisoners exhibit the BTAE? “Please rate yourself in comparison to the average member of the community on each of the following characteristics on a scale of -5 to +5, 0=average” Examples: Moral; kind to others; trustworthy; honest; dependable; compassionate; generous; law-abiding; self-controlled If BTAE exists what would we see?
The ratings should be higher than 0. μ= 0 if there is no better-than-average effect
64
What is the null and alternative hypothesis for the prisoners exhibiting BTAE example?
Null: The participants do not show the BTAE. That is, the scores from our sample will not be greater than a population whose mean μ, is equal to 0. H0: μ=0. Alternative: The participants show the BTAE. The scores from our sample come from a population whose mean, μ, is greater than 0. H1: μ>0.
65
In a single sample Z test why are population parameters known?
So we can determine the mean of the sampling distribution of the mean and the standard error of the mean from the population parameters.
66
If you don't know the variability of the population of individuals, you can estimate it from..?
You can estimate it from what we do know- scores of the people in the sample. The group of people in the study are considered to be a random sample from a particular population so the variability ought to reflect the variability of that population.
67
How will sample variance compare to the variance of the population?
Variance will generally be slightly smaller than the variance of the population from which it is taken. Variance of the sample is a biased estimate of the population variance. It consistently slightly underestimates the actual variance of the population.
68
What is the difference between s^2 and SD^2?
SD^2 underestimates population variance. s^2 is an unbiased estimate of the population variance by changing the denominator to N-1.
69
What is degrees of freedom (df)? (in basic terms)
The number of scores to vary when estimating a population parameter. It is the denominator of s^2
70
What is degrees of freedom (as an explanation)
If you have 5 scores with no constraints, you can produce whatever 5 numbers you wish and you would have 5 degrees of freedom. HOWEVER if you wanted to produce a set of 5 numbers with a mean of 25 you can produce the first 4 numbers with no restraint but then the last number has to be a specific value. We have four degrees of freedom then.
71
When figuring out the variance what do you first have to know?
You first have to know the mean. If you know the mean and all but one of the scores in the sample, you can figure out the one score you don't know. If we have some number, N, of scores with a pre-specified mean, then we only have N-1 degrees of freedom left.
72
When to used SD vs s.
We use SD or SD^2 for descriptive statistics of a set of data (no plans to do inferential analysis) OR when doing inferential analysis and sigma is known. We use s or s^2 when you are doing inferential analyses and population parameters are unknown. You lose one degree of freedom because you use the mean of the sample to estimate population variance.
73
In hypothesis testing, when population variance is unknown, what’s the best estimate of population variance?
The sample variance
74
Why is SS/N a biased estimate of population variance?
It underestimates population variance.
75
What is the mean of the sampling distribution of the mean for the prisoners BTAE example?
The mean of the population is ) (no BTAE) so therefore (from the central limit theorem) the mean of the sampling distribution of the mean =0.
76
Shape of the sampling distribution when the population variance is unknown.
With an estimated population variance, you have less true info and more room for error. The sampling distribution of means is likely to have more extreme means than in an exact normal distribution. The smaller the sample size, the greater the tendency to have extreme means.
77
When population variance is estimated from the sample variance, the distribution will not be a normal curve what do we use instead of the z table?
We use the t distribution and t table.
78
For the shape of t distribution when sample N is small what happens to the values for s?
It can fluctuate a lot a produce distributions that are platykurtic.
79
What do small values for df mean? Large?
There is an increased probability of extreme means, so the shape of the distribution is flattened (platykurtic) As df increases, the distribution becomes less flat; the peak gets taller and the tails get skinnier, lie closer to the horizontal axis. The fewer degrees of freedom used to estimate σ x-bar the more variable the estimate is.
80
T distributions with varying degrees of freedom relative to Z distribution (3 trends/facts)
1. The t distribution is a family of curves that varies with the number of df. 2. For each df, there is only one curve. 3. The smaller the df, the greater the tendency to have extreme means. (eg. t(2) will be more platykurtic and t(60) will have a more normal distribution.
81
When is is it especially important for you to have a more extreme sample mean in order to reject the null?
When you have a smaller sample size (N) and therefore a smaller df. The top 5% has to be farther out on a t distribution compared to the z distribution. This is apparent especially whenever you have a smaller df.
82
The difference between the t and normal distribution is greater with 3 degrees of freedom than with 50 degrees of freedom. a)True b)False
True
83
In order to contain 95% of the distribution, you generally have to go out farther from the mean for a a) t distribution b) Normal distribution
t distribution
84
What is the name of the sampling distribution when you were given μ, but not σ?
Sampling distribution of the t statistic when df=____.
85
What is the statistic for analysis for a t-test?
Sample mean, M or X-bar.
86
How do you determine t(crit) from the Table of critical values?
Based on df and alpha
87
What is t(obs)= to in words?
t= (Statistic-hypothesized value)/Estimated standard error of the statistic.
88
Can we get the actual p-value for t(obs) from the t table?
Can't get actual p-value from table, need to compare t(obs) to t(crit) to make a statistical decision.
89
What is important to note when writing up the formal report after using the t test?
's' is denoted as 'SD' in formal report even though you use the N-1 formula. Need to include Mean and SD of sample, mean of the population (or comparison), the test statistic t, the degrees of freedom in parentheses, and p(obs), or as close to p(obs) as you can get.
90
What is the p(obs) when looking at the SPSS output? What is s x-bar?
p(obs) for a two tailed test= Sig. (2-tailed) ** If you do a one tailed test you need to divide this number by 2) s x-bar= standard error mean

Psyc 300A (4 decks)

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