Module 2

Question 1

What is a case-control study?

Answer 1

A study comparing cases and controls

Question 2

What is a retrospective case control study?

Answer 2

Researchers looking back on how subjects behaved over time, looking at case group and control gorup

Question 3

Draw the casual model(i.e. the “directed acyclic graph). Include the mediator, and have arrows which show “what we already know”, “what we can prove”, and “what we want to know”

Answer 3

Refer to PCV 2.1

Question 4

What is sampling variability/error?

Answer 4

When you draw a random and representative sample from a population, it is not always going to be the exact same sample every time you draw it.

Question 5

3 students draw a sample of 5 observations from the population. How many samples did each student draw from the population? What is the sample size used in the experiment?

Answer 5

1 sample with a sample size of 5

Question 6

What is a sampling distribution?

Answer 6

First you take several samples and take the mean of each sample. Then, you treat the sample means as the new data set and plot a histogram.

Question 7

What is a sample distribution?

Answer 7

A sample distribution would be a histogram of the values within one sample

Question 8

As you increase the sample size, what happens to the standard deviation, graph shape, and mean of the sampling distribution graph for a NORMAL distribution?

Answer 8

The graph tightens, variability decreases, standard deviation also decreases. The shape of the graph does not change.

The mean of the means does not change with an increased sample size.

Question 9

What is n?

Answer 9

The sample size

Question 10

As you increase the sample size, what happens to the standard deviation, graph shape, and mean of the sampling distribution graph for a UNIFORM distribution?

Answer 10

Mean of the means is unchanged

The standard deviation gets smaller

The shape of the sampling distribution becomes more and more normal

Question 11

What are some synonyms for a normal graph

Answer 11

symmetric
gaussian
bell shaped

Question 12

What is the central limit theorem?

Answer 12

When n is “sufficiently” large, the sampling distribution for a particular statistic(e.g. sample mean) will tend towards a normal distribution even if the underlying population distribution is not Gaussian.

Question 13

What is the normal distribution? What are the two parameters for a normal distribution? If a random variable X is distributed normally, then we denote it as ….

Answer 13

The normal distribution is a continuous probability distribution for real-valued random variable. The two parameters for a normal distribution are the mean (mu) and variance(sigma squared).
(Notation in notes)

Question 14

How does the normal distribution differ from the binomial distribution?

Answer 14

The binomial distribution is a discrete probability distribution. This means that the values that our random variable X could take on were clearly delineated integers, like the number of heads in five coin flips. X could not be a fraction like three and a half heads.

In comparison, the values that the random variable X could take on in a normal distribution could include fractions of a whole unit

Question 15

In a normal distribution, what do the mean and variance tell us about the graph?

Answer 15

1. The mean tells you about the location of the distribution. The shape would remain the same if only the mean of a normal distribution is changed
2. The variance tells us about how widely distributed the values are. The centre of the distributions would be the same but the peak of the graph and the width of the graph would change.

Question 16

The probability of observing random, normally-distributed values within a given range is equal to

Answer 16

the associated area under the curve (AUC)

this works by considering intervals of potential values for X as defined on the x-axis of the p-lot, then calculating the proportion of th total area under the curve that falls within that interval. This would give us the probability of observing a random normal variable from this population with a value in that range.

Question 17

Can you calculate the probability for a single value of X in a normal distribution?

Answer 17

No, when working with continuous distributions like the normal distribution, our probability will always be anchored to an interval.

Question 18

If a random variable X is distributed normally, why do we know about the standard deviations?

Answer 18

1. There is an approximately 68% chance X falls within one standard deviation of the mean
2. there is an approximately 95% chance falls within two standard deviations of the mean
3. There is an approximately a 99.7% chance X falls within three standard deviations of the mean

Question 19

The value you get for the probability density function is not a probability in and of itself. What do you have to do to calculate the actual probability?

Answer 19

Use calculus to integrate the PDF across the desired rand and calculate the AUC which tells you the probability X falls in the range.

Question 20

What is a transformation?

Answer 20

transformations are just functions that map a value in one space to a value in a second space. Usually we can identify functions to “back-transform” the new data to the original space(useful transformations will allow us to do this).

Question 21

What is the log transformation look like(i.e. the calculations)? What are log transformations useful for?

Answer 21

To go from original data to log transformed data: take the log of the x value you are trying to transform, keep the y value the same. TO back transform rats the x value to the power of 10.

Log transformation are useful fro mapping right-skewed distributions into more normal distributions in the transformed space.

Question 22

What is the formula for calculating the 95% confidence interval for a population mean? Define each variable.

Answer 22

Refer to notes page

Question 23

Is the confidence interval random?

Answer 23

Yes. Our X bar could be different depending on the specific random sample we take. Moreover, the confidence interval will either cover or not cover the true mean

Question 24

What is the coverage probability?

Answer 24

The franction of samples that are taken from the dataset that over the true population mean

Question 25

What is the difference between mu and X bar?

Answer 25

mu is the population mean

X bar is the sample mean

Question 26

What is one reason why a CI might have a coverage possible lower than 95%? What are two ways this can be fixed?

Answer 26

If the population data is non gaussian(i.e. very skewed), the our confidence interval is not going to have 95% coverage probability. We can fix this by increasing the sample size taken from the dataset. You can also use a log transformation to create a Gaussian distribution, even with a small sample size

Question 27

How well the data conforms to the stated coverage probability depends on the ________(1)

Answer 27

(1) shape of the population distribution and the sample size

Question 28

The standard deviation of the sampling distribution of X bar is known as ______(1). What is the formula for this?

Answer 28

the standard error of X bar

formula in notes

Question 29

What are the three differences between a distribution of gaussian observations vs a distribution of sample means?

Answer 29

Distribution of Gaussian observations:
- made up of individual observations from the populaiton
- Centered at population mean mu
- Variability quantified by standard deviation

Distribution of sample means
- Made up of sample means calculated from infinite samples of size n from the popualtion
- Centered at population mean mu
- Variability quantified by standard error

Question 30

When calculating confidence intervals, we assume that the true value of mu is equal to

Answer 30

X bar

Question 31

How do you calculate the margin of error? How do you calculate the width of the 95% confidence interval?

Answer 31

Refer to notes

Question 32

As n increases, the width of the 95% confidence interval goes _____(1).

Answer 32

(1) down

Question 33

What effect would lowering the standard deviation have on the standard error, the margin of error, and the width of the confidence interval?

Answer 33

It would lower all of them

Question 34

What is the formula for calculating sample mean and sample variance?

Answer 34

Refer to notes sheet

Question 35

What are the three rules that are relevant to log transformed data?

Answer 35

1. The mean of the logged data is almost equal to the median of the logged data
2. The log of the median(of the regular data) is equal to the median of the logged data. This is because the median is an observable value in the dataset
3. The log of the mean(of the regular data) is NOT equal to the mean of the logged data.

Question 36

When you do 10 raised to the power of the mean of the log data. what do you get?

Answer 36

the median of the raw data

Question 37

Mathematically write out rule 1 and rule 2 relating to log transformed data.

Answer 37

Refer to notes `

Question 38

You just calculated the confidence interval for the mean of the data in log dollars. If you exponentiate these values, what would the new interval tell you?

Answer 38

The confidence interval for the median of the raw data

Question 39

What does stratified mean?

Answer 39

Stratification is the process of dividing members of the population into homogeneous subgroups before sampling.

Question 40

What is a histogram?

Answer 40

Graphs with bus that count values in dataset and give us an overview of the data distribution

Question 41

What are the drawbacks of a histogram?

Answer 41

1. Hard to see the centre of distribution to compare “typical” values
2. Hard to plot the two distributions together on the same graph

Question 42

What is a density plot? What is the drawback of a density plot?

Answer 42

SImilair to a historgram but you have lines instead of bars.

It is hard to see the centre of distribution to compare typical values