Home » 国际课程 » A-level课程 » Details

CIE A Level Maths: Probability & Statistics 2复习笔记1.2.3 Confidence Intervals

Category: A-level课程, 教材笔记, 福利干货 Date: 2022年8月18日下午3:08

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

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

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

The z-values for common confidence levels are:
- - z-values for most confidence intervals you will need to work with will be given in the table of critical values

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

Always check whether the population follows a normal distribution, if it does not then you will have to state that the Central Limit Theorem is being used (as the sample size should be big enough). Take care with the fiddly bits:
- You need to square-root the sample size
- You need to use the standard deviation so you might also need to square-root the variance

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

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

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

Misty selects 50 fish at random from a lake. 17 of the 50 fish are trout.

Calculate a 90% confidence interval for the proportion of the fish in the lake that are trout.

1-2-3-confidence-interval-for-p-we-solution

Remember that the confidence interval is not guaranteed to contain the true population proportion. When interpreting your answers use phrases like “there is evidence to support...”.

转载自savemyexams