models, populations and estimations

Question 1

What is a Bernoulli distribution

Answer 1

A Bernoulli distribution is distribution over a binary variable
(which can always be written as {0, 1}).
I It has a single parameter, which we will denote by θ, which gives
the probability that it takes the value of 1

Question 2

What is a normal distribution?

Answer 2

The normal distribution is a probability distribution over a
continuous variable.
I It has two parameters: The mean, usually denoted by µ, and the
standard deviation, usually denoted by σ

Question 3

Populations and samples and inference

Answer 3

• Our statistical models have parameters that are assumed to have
  fixed but unknown values.
• We must estimate these values from data.
• However, our estimates will always be subject to uncertainty.
• To introduce this major topic of statistical inference, we must first
  consider the topic of populations and samples.

Statistics of samples can be used to estimate the true values of
populations.
- But statistics of samples can be very variable.
- But understanding how the statistics vary (i.e. knowing the
sampling distribution), we can
- For example, we can say things like (informally speaking) If the
true mean is 100, the mean of a sample of 10 values could be anywhere
from around 90 to around 110. . . .
- With reasoning like this, we did get a sample of mean of 105, we
could ask (informally speaking), if that is compatible or not with
the true mean being, say, 100

Question 4

populations and samples

Answer 4

The concept of a population is a very important concept in
statistics.
-The population is a (possibly hypothetical, usually infinite) set
from which our data is assumed to be a sample.
- The statistical model is in fact a model of the population: it is a
model of the set from which our data is a sample.
- To understand how we infer the properties of the model from
data, we must understand how samples relate to populations.

Question 5

Sampling distributions

Answer 5

We actually know that IQ in the population is normally
distributed with a mean of 100 and a standard deviation of 15. It
is designed that way.

• But what if we didn’t know that and we were trying infer the
  mean and the standard deviation of IQ in the population from a
  sample.
• We get a sample, calculate the mean and the standard deviation.
• What similar will this mean and standard deviation of this sample
  be to the true mean and standard deviation?

Question 6

r studio for a bell shaped curve

Answer 6

plot_normal(mean = 0, sd = 1)

Plot_normal(mean = 100, sd = 15, xmin = 50, xmax = 150)

Plot_normal_sample(1000, mean = sd = 15, bins = 50)
^^histogram

Question 7

r studio for statistical properties of normal distribution

Answer 7

normal_percentiles(mean = 0, sd = 1)

Normal_percentiles(mean = 100, sd = 15)

Question 8

to create scattergram

Answer 8

Plot_repeat_normal_samples(N = 100, n = 10 mean = 100, sd = 15)

Question 9

to create histogram

Answer 9

hist_repeat_normal_samples(N = 1000, n = 10, mean = 100, sd = 15)

Question 10

samples from normal distribution

Answer 10

Plot_normal_sample(1000, mean = sd = 15, bins = 50

Question 11

repeat normal samples

Answer 11

Plot_repeat_normal_samples(N = 100, n = 10 mean = 100, sd = 15)

Question 12

Sampling distributions from normal distributions to create histogram

Answer 12

hist_repeat_normal_samples(N = 1000, n = 10, mean = 100, sd = 15)