Probability chapter 9

Question 1

Simple random sampling

Answer 1

Every member is equally likely to be chosen. EG: allocate each member of population a no. then use random numbers to chose desired sample.(sequences, )

Question 2

Systematic sampling

Answer 2

Find sample size for n from population N taking 1 number from the first k members of population at random. pick every Kth member where k=N/n

Question 3

Example of systematic sampling

Answer 3

suppose you want to sample 8 houses from a street of 120 houses. 120/8=15, so every 15th house is chosen after a random starting point between 1 and 15.

Question 4

Stratified sampling

Answer 4

(When you want distinct groups in sample) Split groups up into distinct groups + then sample within each group in proportion to its size.

Question 5

Example of stratified sampling

Answer 5

One might divide a sample of adults into subgroups by age, like 18-29, 30-39, 40-49, 50-59, and 60 and above

Question 6

Opportunity sampling

Answer 6

Take samples from members of the population you have access to until you have sample of the desired size.
EG: An example would be selecting a sample of students from those coming out of the library.

Question 7

Quota sampling

Answer 7

Want distinct groups to be represented in sample, decide how many members of each groups you wish to sample in advance + use opportunity sampling until there’s enough of the sample for each group

Question 8

An example of quota sampling

Answer 8

A researcher might ask for a sample of 100 females, or 100 individuals between the ages of 20-30.

Question 9

Cluster sampling

Answer 9

Split the population into clusters that you expect to be similar to each other, the take a sample from each of these clusters.

Question 10

Example of cluster sampling

Answer 10

For example, a researcher wants to survey academic performance of high school students in Spain. He can divide the entire population (population of Spain) into different clusters (cities). Then the researcher selects a number of clusters depending on his research through simple or systematic random sampling.

Question 11

When deciding on a sampling method

Answer 11

1) Consider whether or not you can list every member of the population.
2) Identify any sources of bias + any difficulties you might face in taking certain examples.
3) Compare the different sampling methods you have available + choose that one that best suits your needs and limitations.

Question 12

When is a sampling method biased?

Answer 12

If it creates a sample that does not represent the whole population .

Question 13

To solve a problem about summary statistics

Answer 13

1) Identify the summary statistics appropriate to the problem.
2) Calculate the values of the required statistics, using a calculator where appropriate.
3) Use the statistics to describe key features of the data set and make comparisons.
4) If not already done, identify any outliers and remove them, then see how this affects calculations.

Question 14

Define outliers.

Answer 14

Are values that lie significantly outside the typical set of values of the variables.

Question 15

Mode

Answer 15

PROS:
-Useful for non-numerical data
-Not usually affected by outliers
-Not usually affected by errors or omissions
-Is an always observed data point
CONS:
- Doesn’t use all of the data
-May not be representative if it has a low frequency
-There may be other values with similar frequency

Question 16

Median

Answer 16

PROS:
- Not affected by any outliers
-Not significantly affected by errors
CONS:
-Doesn't make use of all the data

Question 17

Mean

Answer 17

PROS:
- When the data set is large a few extreme values have negligible impact.
CONS:
-When data set is small a few extreme values have a big impact

Question 18

Range

Answer 18

PROS:
Reflects the full data set
CONS:
Distorted (misrepresented) by outliers

Question 19

IQR

Answer 19

PROS:
Not distorted by outliers
CONS:
Doesn’t reflect all of the data

Question 20

Standard deviation

Answer 20

PROS:
When the data set is very large, few outliers have negligible impact.
CONS:
When the data set is small few outliers have a big impact.

Question 21

distribution

Answer 21

How often each outcome occurs. Each outcome with a given frequency

Question 22

continuity correction

Answer 22

Involves altering end points of an interval of rounded data to include values which would fall in the interval when rounded.

Question 23

Frequency density

Answer 23

= frequency / class width

Question 24

Histogram

Answer 24

You can use a histogram to display continuous data

Question 25

When deciding on the appropriate diagram to represent data

Answer 25

1) Consider whether you need to be able to display all of the values, including outliers.
2) Consider whether you need to be able display relative or absolute frequencies.
3) Consider whether you are more interested in displaying the distribution or the summary statistics.

Question 26

Box plot

Answer 26

Advantages:
Highlights outliers.
Makes it easy to compare data sets.
CONS:
Data is grouped into 4 categories so detailed analysis is not possible

Question 27

Histogram

Answer 27

PRO:
Clearly shows shape of distribution
CONS:
Doesn't always highlight outliers
It is possible but not easy to estimate Q1, Q2 and Q3

Question 28

Cumulative frequency curve

Answer 28

PRO:
Makes it easy to find the 5 number summary
CON:
Doesn’t always highlight outliers
If interval boundaries are not shown the degree of detail is not clear.

Question 29

When interpreting a diagram displaying data

Answer 29

1) Consider what is being represented + whether your data is discrete or continuous.
2) If necessary, identify any outliers or missing/ incorrect data, and consider the effects of removing them.
3) Read what is being asked for in the question and use the diagram to answer.

Question 30

What are variables that are statistically related?

Answer 30

They are describes as correlated. There are 3 types of positive r=1 , negative r=-1 + zero r=0

Question 31

To solve a question about bivariate data + correlation

Answer 31

1) Draw a scatter diagram to identify any correlation between 2 variables.
2) Identify data points that don’t fit the general pattern shown by the data
3) Use correlation in the scatter diagram to determine the value of the missing data points.