Statistical hypothesis testing - Topic 5

Question 1

Year 1 - Chapter 7.1

What hypotheses are needed for a test?

Answer 1

• The null hypothesis, H₀, is the hypothesis that you assume to be correct
• The alternative hypthesis, H₁, tells you about the parameter if your assumption is shown to be wrong

If the likelihood is less than a given threshold, called the significance level of the test, then you reject the null hypothesis.

Question 2

Year 1 - Chapter 7.2

What is the critical region and how do you find it?

Answer 2

A critical region is a region of the probability distribution which, if the test statistic falls within it, would cause you to reject the null hypothesis. The critical value is the first value to fall within the region. The critical value and the region can be determined from binomial distribution tables, or by finding cumulative binomial probabilities using your calculator. For a two-tailed test, there would be two critical regions on either end of the distribution.

Question 3

Year 1 - Chapter 7.2

What is the actual significance level?

Answer 3

It is the probability of incorrectly rejecting the null hypothesis.

Question 4

Year 1 - Chapter 7.3

How do you carry out a one-tailed hypothesis test?

Example:

Answer 4

X~B(20,0.4)

• H₀: ρ = 0.4, H₁: ρ > 0.4
• P(X ≥ 11) = 1 - P(X ≤ 10)
• = 1 - 0.8725
• = 0.1275 = 12.75%
• 12.75% > 5%, therefore there isn’t enough evidence to reject H₀

Question 5

How do you carry out a two-tailed hypothesis test?

Answer 5

Carrying out a two-tailed test is the same as one-tailed, however, halve the significance level to give the associated probability for one tail. Then, double your calculated probability to obtain the p-value, and compare this with the significance level of the test.

Question 6

Year 2 - Chapter 1

Relationship for the exponential model

(Its equation)

Answer 6

y = kb^x

ꜜ
Y = log(y); X = x

ꜜ
log(y) = log(k) + x(log(b))

ꜜ
Y = log(k) + X(log(b))

Question 7

Year 2 - Chapter 1

Relationship for the geometric model

(Its equation)

Answer 7

y = ax^n

ꜜ
Y = log(y); X = logx

ꜜ
log(y) = log(a) + n(log(x))

ꜜ
Y = log(a) + nX

Question 8

Year 2 - Chapter 1

What does PMCC measure?

Full definition

Answer 8

The product moment correlation coefficent describes the linear correlation between two variables. It can take values between -1 and 1.

r = 1, perfect positive linear correlation
r = -1, perfect negative linear correlation
r = 0, no linear correlation

Calculated on the calculator

Question 9

Year 2 - Chapter 1

How to hypothesis test for correlation?

For both one-tailed and two-tailed tests

Answer 9

1. Establish your hypothesis;
   H₀: ρ = 0,
   H₁: ρ > 0 (for positive)
   H₁: ρ < 0 (for negative)
   H₁: ρ ≠ 0 (for two-tailed)
2. Calculate the PMCC of the sample to get r
3. Find the critical value in the correct significance level column from the correlation table in the examboard-issued formula book

(Correlation is symmetrical = for negative correlation, put minus sign in front of number)

Question 10

Year 2 - Chapter 3.7

How do you carry out a hypothesis test for normal distribution?

Example:

Answer 10

X̅~N(μ, σ²/n), where n is the sample size taken at random.

• X~N(60, 3²)
• H₀: μ = 60, H₁: μ < 60
• X̅~N(60, 3²/16)
• P(X̅ < 59.1) = 0.1151
• 0.1151 > 0.055, therefore there’s insufficient evidence to reject H₀

Question 11

Year 2 - Chapter 3.7

How do you find the critical region or value for a normal distribution?

Answer 11

You need the sample mean of the normally distributed variable.

Z = (X̅ - μ)/(σ/√n), which is a normall distributed random variable with Z~N(0,1²)

You can also you the inverse normal distribution function in your calculator to directly work out X̅.