S2

Question 1

What is the sum of the probabilities in a probability distribution

Answer 1

1

Question 2

How do you calculate the mean and variance of a cumulative distribution function

Answer 2

1. Mean= total (x1*p1) Multiply the x-values by the probabilities
2. Variance= total(X1^2*P1)-mean^2

Question 3

List the variation for the mean and variance for (aX), (x+b) and (aX+b)

Answer 3

1. E(ax) = aE(X)
2. E(X+b) = E(X) + b
3. E(aX+b) = aE(X)+b
4. Var(aX)= a^2Var(X)
5. Var(X+b) = Var(X)
6. Var(aX+b) = a^2Var(X)

Question 4

When is poisson distribution used

Answer 4

1. When events occur independently
2. At random
3. At a constant average rate

Question 5

State 4 things about probability density function f(x)

Answer 5

1. Area under the graph represents the probability
2. Total area under the curve=1
3. If linear graph use formula for area of triangle or trapezium
4. If quadratic- integrate to calculate probability

Question 6

How do you find E(x) and Var(x)of probability density function f(x)

Answer 6

1. E(x)= integral( f(x)*x) dx

2. Var(x)= integral( f(x)*x^2) dx - [E(x)]^2

Question 7

How do you find E(x) and Var(x) of a rectangular distribution

Answer 7

1. E(x) = 1/2(a+b)

2. Var(x) = 1/12 (b-a)^2

Question 8

How would you find E(1/x) for a probability density function

Answer 8

1. Integral ( f(x) *1/x) dx

Question 9

What is the cumulative distribution function F(x)

Answer 9

1. Integral of f(x)

2. Useful when finding medians F(X)=0.5, lower quartile F(x)=0.25

Question 10

What are the different scenarios that would result in using z or t in hypothesis testing

Answer 10

1. Population variance given- Any sample size- use z
2. Population variance unknown- large sample size >30- use z - central limit theorem
3. Population variance unknown- small sample size <30 - use t

Question 11

What is the formula for degrees of freedom

Answer 11

1. d.f=n-1

Question 12

What is the formula for confidence intervals

Answer 12

1. x̅ +/- (z or t) * Root(Population variance/n)

Question 13

What may be the different assumptions when doing confidence intervals

Answer 13

1. The data can be regarded as a random sample
2. If large sample size- central limit theorem
3. Contents are normally distributed if small sample size with t

Question 14

What are the steps carried out in hypothesis testing

Answer 14

1. State null hypothesis H0:μ=a
2. State the alternative
   H1 : μ =≠ a - two-tailed
   H1:μa- One-tailed positive critical value
3. Test statistic z or t
   Z/t= (x̅-μ)/root(population variance/n)
4. Use calculator to find the critical value
5. If the value of the test statistic falls in the critical region then you reject H0 and it is significant

Question 15

What is the formula for expected frequency for Chi-squared

Answer 15

1. Row total * column total / total

Question 16

What is the formula for the normal test statistic

Answer 16

1. Total = (O-E)^2/ E
2. O: observed frequency
3. E: expected frequency

Question 17

What are two special cases in Chi-squared

Answer 17

1. 1 degree of freedom- Use Yates’ correction

2. The expected frequency must be bigger than 5- if not group the columns appropriately

Question 18

What is the formula for Yates’ correction

Answer 18

1. Total (mod(O-E)-0.5)^2/E

Question 19

When do you accept H0 using Yates correction

Answer 19

1. When Chi data < Chi n

2. When the value you worked out from the data is smaller than the calculator value.

Question 20

What is a type 1 error in hypothesis testing

Answer 20

Rejecting H0 and accepting H1 when H0 is actually correct

Question 21

What is a type 2 error in hypothesis testing

Answer 21

H0 is accepted even though it is incorrect

Question 22

How do you find the variance from the mean in poisson distribution

Answer 22

Mean=variance