Confidence Intervals

Question 1

Which two things do we need to estimate a confidence interval?

Answer 1

The mean and the standard error.

Question 2

What do confidence intervals tell us?

Answer 2

How confident we can be that our mean or our proportion fall within a certain range.

Question 3

Write the formula for finding the 95% confidence interval.

Answer 3

mean + or - (1.96 * standard error)

Question 4

Write the formula for finding the 99% confidence interval.

Answer 4

mean + or - (2.58 * standard error)

Question 5

What is the difference between finding confidence intervals for the mean?

Answer 5

We look at t-distributions and t-scores instead of normal distributions and z-scores.

Question 6

What is the case for if we have a sample size above 100 when calculating the confidence interval of the mean?

Answer 6

The t-distribution is identical to the normal distribution.

Question 7

How does sample size affect the confidence interval?

Answer 7

As the sample size gets larger, the confidence interval gets narrower.

Question 8

How is it finding the confidence interval of the mean when the sample size is small?

Answer 8

As the sample size decreases from 100, the t-distribution gets flatter.

Question 9

How do we estimate t-scores as the sample size changes?

Answer 9

Using Degrees of Freedom

Question 10

How do we find the Degree of Freedom for small sample sizes?

Answer 10

DF = n - 1

Question 11

Why do we subtract 1 from the sample size to calculate the Degree of Freedom?

Answer 11

To recognise that there will be some error in estimating our t-score.

Question 12

How is significance determined?

Answer 12

Based on how our predicted values compare to the values we find in our data.

Question 13

What is a hypothesis?

Answer 13

A clear statement of a relationship, or a statement of a predicted value or range of values.

Question 14

Which 4 concepts give an insight into whether a value is significant or not?

Answer 14

• The null hypothesis
• The alternative hypothesis
• The alpha (α) value or alpha level
• The p-value

Question 15

What does statistical significance suggest?

Answer 15

A relationship (difference) we observe is genuine, i.e. can be found in population, rather than random chance.

Question 16

What is the first type of error in hypothesis testing?

Answer 16

The incorrect rejection of the null hypothesis.

Question 17

Give 2 features about the incorrect rejection of the null hypothesis.

Answer 17

• Chance that this happens is known as significance level or alpha level
• Is not affected by sample size as it is set in advance

Question 18

What is the second type of error in hypothesis testing?

Answer 18

• The incorrect acceptance of the null hypothesis.

Question 19

Give 3 features about the incorrect acceptance of the null hypothesis.

Answer 19

• Probability is beta
• Beta depends upon sample size and alpha level
• Beta gets smaller as the sample size gets larger

Question 20

What is usually the maximum α-level?

Answer 20

0.05

Question 21

What does a α-level of 0.05 mean?

Answer 21

It’s less than 5% likely that our relationship occurred through chance alone.

Question 22

List the 4 steps of generating a hypothesis test.

Answer 22

Step 1: State H(0) and H(a) with clear reference to the population
Step 2: Choose the level of Type I error to be tolerated (i.e. choose the significance or alpha level)
Step 3: Estimate how likely it is that we would observe the sample outcome, e.g. a mean or proportion, if H(0) were true
Step 4: Make a decision. If we are “very unlikely” to have observed a sample estimate under the assumption that H(0) is true, we reject H(0) in favour of H(A)

Question 23

List the 3 different approaches to carrying out a hypothesis test.

Answer 23

• Confidence interval approach
• Test statistic approach
• p-value approach

Question 24

Which steps do the approaches affect?

Answer 24

3 & 4

Question 25

What are steps 3&4 of carrying out a hypothesis test with the confidence interval approach?

Answer 25

Consider if the value of μ is assumed under H(0) inside the 95% confidence interval for μ.

Question 26

What are steps 3&4 of carrying out a hypothesis test with the test statistic approach?

Answer 26

We convert the sample mean to a Z-score.
e.g. x̄=28, μ=25 and SE = 8/√30 = 1.46
Z = 28 – 25 / 1.46 = 2.05

Question 27

Give the two possible outcomes of Z associated with a critical region.

Answer 27

Z in critical region (i.e. Z ≤ -1.96 or Z ≥ + 1.96): reject H(0)
Z not in critical region (i.e. -1.96 and + 1.96): we do not reject H(0)

Question 28

What does Z being in the critical region mean?

Answer 28

The value of our sample mean is so far away from the hypothesised value of the population meaning that there is less than 5% probability that it could have occurred by chance, were H(0) true.

Question 29

What are steps 3&4 of carrying out a hypothesis test with the p-value approach?

Answer 29

If the p-value (probability of getting a test statistic greater or equal to the value) is very small, this mean that we are very unlikely to have got the sample mean as extreme as 28 if H(0) were true. So we reject H(0).

Question 30

What is considered by a small p-value?

Answer 30

Determined by our chosen significance or alpha level (e.g. 5% or 0.05).