1.2.3 Confidence Intervals

What is a confidence interval?

• It is impossible to find the exact value of a population parameter when taking a sample
• The best we can do is find an interval for which the exact value is likely to lie within
  • This is called a confidence interval
• The confidence level of a confidence interval is the probability that the interval contains the population parameter
  • Be careful with wording – the population parameter is a fixed value so it does not make sense to talk about the probability that it lies within an interval
    • Instead we talk about the probability an interval contains the parameter
  • Suppose samples were collected and a 95% confidence interval for a population parameter was constructed for each sample, then for every 100 intervals we would expect on average 95 of them to contain the parameter
    • 95 out of 100 is not guaranteed – it is possible all of them could contain the parameter
    • It is possible (though very unlikely) that none of them contains the parameter

What affects the width of a confidence interval?

• The width of a confidence interval is the range of the values in the interval
• The confidence level affects the width
  • Increasing the confidence level will increase the width
  • Decreasing the confidence level will decrease the width
• The size of the sample affects the width
  • Increasing the sample size will decrease the width
  • Decreasing the sample size will increase the width

How do I calculate a confidence interval for the population mean (μ)?

• For this course we only construct symmetrical confidence intervals for the mean of a population when:
  • The variance of the population is known
  • The population follows a normal distribution
  • The population does not follow a normal distribution but the sample size is big enough to use the Central Limit Theorem
• The confidence interval can be found using the formula

• n is the size of the sample
• σ is the standard deviation of the population
• is the mean of the sample
•  is the value such that where α is the confidence level

• The width of the confidence interval is

• Note that the sample mean is the midpoint of the confidence interval and does not affect the width

How do I find the z-value for a confidence interval?

• You use the standard normal distribution 
• If the confidence level is % we find such that %
• To do this find - it can be shown that this is %

• The z-values for common confidence levels are:
  • For 90%:  = 0.95  so z = 1.645
  • For 95%: = 0.975 so z = 1.960
  • For 99%:  = 0.995 so z = 2.576
    • z-values for most confidence intervals you will need to work with will be given in the table of critical values

How can I interpret a confidence interval?

• After you have calculated a confidence interval for μ you might be expected to comment on the possibility of μ being a specific value
• If the value which is claimed to be μ is within the confidence interval then there is evidence to support the claim
• If the value is outside the interval then there is not enough evidence to support the claim

How do I calculate a confidence interval for the proportion (p) of a population?

• If we want to find estimate the proportion of a population that fulfil a certain criteria we can construct a confidence interval based on the proportion of a sample who fulfil that criteria
  • The proportion is between 0 and 1
• For this course we only construct symmetrical confidence intervals for the proportion of a population provided that the sample size is large enough to use the Central Limit Theorem
• The confidence interval can be found using the formula

• n is the size of the sample
• (or p_s) is the proportion of the sample
• z is the value such that where  is the confidence level

• In the formula booklet you are given the distribution of the sample proportion which might help you remember the formula for the confidence interval

• The width of the confidence interval is

• Note that the sample proportion is the midpoint of the confidence interval but it also affects the width

How can I interpret a confidence interval?

• Interpreting a confidence interval for p works in the same way as a confidence interval for μ
• The only additional part is that you might get asked to see whether an experiment is fair
  • Find the probability of the outcome as though it were fair
    • For example – a fair coin will have a 0.5 chance of landing on each side
  • Use a sample to calculate a confidence interval
  • See if the probability is in the confidence interval
    • If it is not then there is sufficient evidence to suggest that the experiment is not fair