05 Confidence Intervals

Question 1

What is a point estimate?

Answer 1

A single value used to estimate a population parameter, such as a sample mean x̄ used to estimate μ, or a sample proportion p̂ used to estimate p

Question 2

What are the best estimates of μ and σ?

Answer 2

x̄ is the best estimate of μ, and p̂ is the best estimate of p

Question 3

What is a confidence interval?

Answer 3

A range of values that the true population parameter (μ or p) is likely to lie in

Question 4

What is a confidence level, C?

Answer 4

The probability that the population parameter lies within the confidence interval

Question 5

What is a margin of error, ME?

Answer 5

Statistic ± ME = the confidence interval for the parameter

Question 6

How do we calculate ME?

Answer 6

• ME = (critical value)⋅(standard deviation of statistic)
• Critical values are found from the t-distribution table
• Formulæ for the standard deviations of the statistics are on the formula sheet

Question 7

All the methods for calculating confidence intervals are only valid if…

Answer 7

the samples are random

Question 8

How large is a large sample?

Answer 8

n ≥ 30

Question 9

When can we use the normal distribution z-scores as critical values in confidence intervals for means?

Answer 9

If the population is normal or the sample is large

Question 10

We don’t usually know σ, so….

Answer 10

We usually use s to estimate it

Question 11

What are the steps to calculating a confidence interval?

Answer 11

1. Write down the relevant values from the question
2. Verify that the relevant conditions and assumptions are met
3. Find the critical value from the tables
4. Calculate the ME and confidence interval
5. Write down what it means in the context of the question you are answering

Question 12

How do you find the z_c value if C is not on the bottom of the t-distribution table?

Answer 12

Sketch the normal distribution curve, and use the tables to find z_c such that C of the data lies between ±z_c

Question 13

How do you find critical values for a small sample?

Answer 13

Use the t-distribution

Question 14

How do you look up values on the t-distribution table?

Answer 14

By looking for degrees of freedom, df

Question 15

What are degrees of freedom, df?

Answer 15

• The number of independent observations
• df = n-1

Question 16

What happens to the t-distribution as df increases?

Answer 16

• the variability of the curve falls (it gets narrower in shape)
• its shape approaches the shape of the normal distribution
• when df = 29, the difference between the two is very small, so when n ≥ 30 we can use the normal distribution

Question 17

What are the conditions/assumptions to verify to calculate confidence intervals for proportions?

Answer 17

1. the sample is random
2. N > 10n
3. np > 10, nq > 10

Question 18

What are the conditions/assumptions to be verified to work out confidence intervals for the difference between two means?

Answer 18

1. the samples must be random
2. the samples must be independent
3. the population must be normal or n₁ + n₂ ≥ 40

Question 19

How should you decide whether to use the t-distribution or normal distribution to calculate a confidence interval for the difference between two means?

Answer 19

• df is the smaller of n₁-1 and n₂-1
• if df ≤ 29, use the t-distribution
• if df > 29, use the normal distribution

Question 20

What are the condtions/assumptions to be verified to calculate the difference between two proportions?

Answer 20

1. the samples are random
2. the samples are independent
3. N > 10n for both samples
4. np̂ > 10 and nq̂ > 10 for both samples

Question 21

Why is there a minimum sample size n necessary to get a usable confidence interval for any given confidence level C?

Answer 21

Confidence intervals which are too wide are useless, and the size of the interval narrows as n increases. The maximum useful confidence interval width therefore corresponds to a minimum necessary sample size n.

Question 22

How can you calculate the minimum necessary sample size?

Answer 22

Rearrange the expression given for ME on the formula sheet to give an expression for n in terms of z_c, ME, and σ, and then solve for n.

Question 23

When calculating the minimum sample size n necessary, we do not know σ. What are the three common ways to estimate it?

Answer 23

• use values from previous studies
• use a conservative estimate of √(pq/n) with p = 0.5
• use the rule of thumb σ = R/4, where R = the range of values we expect to find in the population

Question 24

What must be remembered when calculating the minimum sample size?

Answer 24

Always round a decimal answer up. It is a minimum, so rounding it down will give a wrong answer