WEEK 3: Limited samples and prediction of the main features of a variable, I

Question 1

What is a population?

Answer 1

a collection of all possible individuals, objects or measurements of interest

Question 2

What is a sample?

Answer 2

a portion or part of the population of interest

Question 3

What is the goal of a sample?

Answer 3

representativeness - necessary if you want to generalise to a population + to allow the use of inferential techniques

Question 4

What is a probability sample?

Answer 4

A sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample

Question 5

What are the criteria / methods to select a representative sample?

Answer 5

sample size
sampling methods
sampling error

Question 6

What 3 factors determine the size of a sample?

Answer 6

the desired level of reliability (Level of Confidence)

the margin of error the researcher will tolerate

the variation in the population being studied

Question 7

How many number of observations is regarded as the minimum for findings to be statistically reliable?

Answer 7

100 observations

Question 8

What four methods can be used to construct a probability sample?

Answer 8

Simple Random Sampling

Systematic Random Sampling

Stratified Random Sampling

Cluster Sampling

Question 9

What is simple random sampling?

Answer 9

every sample of size n has the same probability of being selected - equal chance / included

eg Kent school has 845 students, teacher wants to do study and selects a random sample of 52 students. Each student writes their name on a slip of paper and puts it in box, after being mixed the first selection is made, process is repeated until the sample of 52 students is chosen

process = sampling without replacement, so probability of each selection changes eg 1/845, 1/844, 1/843

Question 10

How would you perform simple random sampling in excel?

Answer 10

Data –> Data Analysis –> Sampling –> Fill boxes with info

eg input range?
random
no of samples
output range?

Question 11

What is systematic random sampling?

Answer 11

items are taken at regular intervals eg every 5th person in a list

items/individuals of the population are arranged in some order, a random starting point is selected and then every k^th member of the population is selected for the sample

Question 12

How do you calculate K? (Systematic Random Sampling)

Answer 12

k = population size / sample size

Question 13

A population of students at Kent school is 845 students, a sample of 52 students will be selected from this, how would you do a systematic random sample?

Answer 13

1) work out K
–> K = population size / sample size = 845/52 = 16

random sampling is used in the selection of the first name among the first 16 students, then select every 16th name on the list after.

Question 14

What is stratified random sampling?

Answer 14

a population is first divided into subgroups (strata) and a sample is selected from each stratum

(the population is divided into appropriate categories + sample is chosen so the proportions in each category are the same as the proportions in the population)

Question 15

Population size = 400
Sample size = 50
Companies are divided into stratum, the number of firms in each stratum is:
1 = 24
2 = 43
3 = 208
4 = 119
5 = 6

How would you do a stratified random sample?

Answer 15

1) work out relative frequency for each company (in table) = number of firms / population size (total should equal 1 when added together
2) work out the number sampled = relative frequency x sample size (when added together, total should equal sample size)

Question 16

What is cluster sampling?

Answer 16

uses a natural division of the population
first chooses a random sample of clusters and then a sample from each cluster - usually contains random sampling

Question 17

What is the sampling error?

Answer 17

the difference between a sample statistic and its corresponding population parameter

sampling error = (sample statistic - population parameter)

Question 18

What is the sample statistics symbol for the mean?

Answer 18

X̄

Question 19

What is the population parameter symbol for the mean?

Answer 19

μ

Question 20

What is the sampling error of the mean?

Answer 20

X̄ - μ

Question 21

What is the sample statistics symbol for the standard deviation?

Answer 21

s

Question 22

What is the population parameter symbol for the standard deviation?

Answer 22

σ

Question 23

What is the sampling error of the standard deviation?

Answer 23

s - σ

Question 24

What is the sample statistics symbol for the Variance?

Answer 24

s^2

Question 25

What is the population parameter symbol for the Variance?

Answer 25

σ^2

Question 26

What is the sampling error of the variance?

Answer 26

s^2 - σ^2

Question 27

What is the sample statistics symbol for the Proportion?

Answer 27

p

Question 28

What is the population parameter symbol for the Proportion?

Answer 28

π

Question 29

What is the sampling error of the proportion?

Answer 29

p - π

Question 30

What is the sampling distribution of the sample mean?

Answer 30

a probability distribution function consisting of all possible sample means of a given sample size size selected from a population

Question 31

What will the mean of the distribution of sample means be exactly equal to?

Answer 31

the population mean

–> the mean of the distribution of sample means = the population mean

Question 32

What is σX̄ ? (the standard deviation of the sampling distribution of the sample mean?)

Answer 32

σX̄ = σ / √n

Question 33

How do you work out the population mean (μ)?

Answer 33

μ = ∑Xi / N

Question 34

How do you work out the population standard deviation (σ)?

Answer 34

σ = √(∑(Xi-μ)^2/N)

Question 35

How would you work this out

“What is the sampling distribution of the sample mean for samples of size 2?”

on excel?

Answer 35

In Excel use the function =COMBIN(7, 2) to calculate the total number of
combinations of size 2 (second argument from a population of 7 items.

We first create all possible samples of size two, using all possible 21 combinations of the original 7 individuals, and then we calculate the mean of each of these 21 samples.

Question 36

Why is sampling preferred to surveying the whole population?

Answer 36

Surveying the whole population can be very expensive.

It can be physically impossible to find all the items in the population.

Sample estimates can be very accurate.