S3

Question 1

How do you find Sxx from a list of numbers

Answer 1

Square all of them add together. Then subtract the mean squared multiplied by the number of numbers.

Question 2

What does PDF stand for?

Answer 2

Probability density function

Question 3

How would you show that a statement is constant with f(x) being a PDF?

Answer 3

Integrate between the range and it will be 1

Negative probabilities won’t occur

Question 4

How do you find the mean of a continuous random variable from the PDF?

Answer 4

∫xf(x) between the given limits. This is expectation of x

Question 5

How do you find the variance of a continuous random variable from the PDF?

Answer 5

∫x²f(x) between the limits
Subtract E(x)²
Var(X)= E(X²)-E(X)² State this explicitly.

Question 6

If you found the mean and variance of a random variable X, what would the mean and variance of Y be if Y=4X-1

Answer 6

Multiply the mean by 4 and subtract 1

Multiply variance by 4²

Question 7

How do you find the probability of a random variable being between two limits when given a PDF

Answer 7

Intergrate between the two limits.

Question 8

How is mode found with a PDF?

Answer 8

It is the highest point.

Differentiate to find it.

Question 9

How is the median found from a PDF?

Answer 9

The integral between m and one of the limits is 0.5

Question 10

What is a uniform distribution?

How is mean and variance found?

Answer 10

Looks rectangular when drawn
Mean is (a+b)/2
Variance = (1/12)(b-a)²

Question 11

If given a PDF of f(x), how would you find the CDF?

Answer 11

The CDF a F(x) is the integral of the PDF between 0 and x

Question 12

What so you have to say when writing down a PDF?

Answer 12

F(x) =………..
Give the limits of the function
Say that it is 0 otherwiise

Question 13

What do you assume when doing a chi squared test?

Answer 13

Observed frequencies are approximately normally distributed about the expected frequencies.

Question 14

When are groups put together with chi ² tests?

Answer 14

To make sure all groups have expected frequencies greater than 5

Question 15

How are the degrees of freedom found with chi² squared test?

Answer 15

= no. of classes - no. of estimated parameters -1

Question 16

What is equal to what with a 🐠 distribution ?

Answer 16

Mean and variance

Question 17

What is assumed for a 🐠 distribution ?

Answer 17

Events take place randomly, independently and at a constant overall mean rate.

Question 18

What do you assume for a binomial distribution

Answer 18

Events are random and independent.

The mean is np

Question 19

What are confidence intervals a measure of?

Answer 19

In repeated sampling

90% of intervals generated in this way would contain the true population mean.

Question 20

How are confidence limits found?

Answer 20

Sample mean ± K*(σ÷√(size of sample))

Question 21

What is the difference between s and σ

Answer 21

S is the sample SD
σ is the population SD
When finding confidence limits you often have to approximate ⍬ with S.
When n≥50 it is a good aproximation.

Question 22

What is the standard error of the mean?

Answer 22

The standard deviation of the sample means is σ÷√n

Question 23

What does the central limit theorem say?

Answer 23

For a sample of size n drawn from a distribution with mean μ & variance σ² the distribution of the sample mean is approximately
N(μ,(σ²÷n)) for sufficiently large values of n.(25+)

Question 24

When do you use 1.645, 1.96, and 2,576 values of K?

Answer 24

When you already know the parent population SD σ
If you estimate σ from the sample, you use the t distribution and not the normal provided the parent population is normally distributed. Thus you use the values of k further up the table.

Question 25

If you had a large sample and you wanted to produce confidence intervals but had to estimate the parent population SD (σ) what values of K would you use?

Answer 25

If n is large enough (50+) then confidence intervals worked out using the normal distribution will be accurate enough and you can use the values of k at the bottom of the table.

Question 26

When conducting a t test with paired data, what is done with significance levels?

Answer 26

When looking for the critical value in the table, look in the column with double the significance level given in the question. If it is a two tailed test, just use the given level.
This is because the tables are made to give each tail a probability of 0.5p%

Question 27

Assumptions for wilcoxen test

Answer 27

The data is symmetrically distributed about the mean/median

Question 28

When conducting a Wilcoxen test, what is done with any data that is equal to the median?

Answer 28

Removed from sample.

Question 29

What are W₊ and W₋?

Answer 29

W₊ sum of the ranks above the median.

W₋ sum of the ranks below the median.

Question 30

How do you check W₋ and W₊ are correct?

Answer 30

W₋+W₊ = 0.5n(n+1)

Question 31

When conducting a Wilcoxen test when is W+ and W- used?

Answer 31

If two tailed the smaller of W- and W+

If one tail test and you are testing if the true median is less than the believed median, use W+

Question 32

Where are the critical values found for the Wilcoxen test?

Answer 32

Page 30 of the formula booklet.

Question 33

For the Wilcoxen test, how do you compare W and the critical value?

Answer 33

If W is less than or equal to critical value, reject H0 and accept H1

Question 34

When do you use the normal distribution?

Answer 34

When the sample is large

When the sample is small, the distribution is normal and the population variance is known.

Question 35

When do you use the Wilcoxen test?

Answer 35

When a sample is small and nothing is known about the distribution of the background population.

Question 36

When do you use the t-distribution ?

Answer 36

The sample is small, the distribution is normal and the population variance isn’t known.

Question 37

What is opportunity sampling?

Answer 37

Sampling which selects from those that are (easily) available

Question 38

If a PDF is f(x)=x/50 for 0 to 10, how would you find E(3x+4)?

Answer 38

Multiply (3x+4) by x/50 for 1/50*(3x² +4x) and integrate between limits.

Question 39

How would you find E(x²)

Answer 39

Multiply the PDF by x² and intergrate between range

Question 40

What is subtracted from what with a Wilcoxen test?

Answer 40

Values - median being tested

Question 41

If when doing a Wilcoxen test and the critical value is equal to the W value, what do you do?

Answer 41

Significant. Accept H1

Question 42

Why is paired data appropriate on occasion?

Answer 42

To remove differences between groups, people, authorities

Question 43

What is the significance level?

Answer 43

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual differe

Question 44

If the mean duration of an event can be normally distributed with ( μ,σ^2), what distribution would give the mean time for three events to occur?

Answer 44

It would be Normally distributed and the mean would be the same.
Variance of total time is 3σ^2, but when you have to divide by 9 to get the variance for the mean.
You divide by 9 and not 3 as 9=3^2 and you have to square constants when multiplying by a variance.

Question 45

Give two reasons why an investigator might need to take a sample in order to obtain information about a population.

Answer 45

For example, need to take a sample because the population might be too large for it to be sensible to take a complete census.
Because the sampling process might be destructive

Question 46

State two requirements of a sample.

Answer 46

Sample should be unbiased
Sample should be representative (of the population)
data should not be distorted by the act of sampling;
data should be relevant.

Question 47

Discuss briefly the advantage of the sampling being random.

Answer 47

A random sample … enables proper statistical inference to be undertaken …… because we know the probability basis on which it has been selected

Question 48

How is the median found from a CDF?

Answer 48

solve the equation F(m) = 0.5

Question 49

In what form is a confidence interval given?

Answer 49

If you found it was 99 to 101 you would write it (99,101)

Question 50

What do you assume when using a t test?

Answer 50

The sample is taken from a normally distributed population.

Question 51

How would you conduct a cluster sample?

Answer 51

Identify clusters which are capable of representing the population as a whole
Choose a random sample of clusters
Randomly sample within the chosen clusters.

Question 52

What is a simple random sample?

Answer 52

A simple random sample is one where every sample of the required size has an equal chance of being chosen.

Question 53

Why might you conduct a cluster sample over a simple random sample ?

Answer 53

If you were investigating school chlidren, simple random would mean talking to a few pupils at lots of different schools which is more work than making a few schools clusters and sampling them.

Question 54

What is stratified sampling and why might you do it?

Answer 54

There are identifiable subgroups or strata that might exhibit different characteristics.
Each stratum is randomly sampled.
Use it to obtain a representative sample.
Can get information on the individual strata.