GEA1000 statistical inferences chapt 4 (confidence intervals)

rahhhhh why stats bro

why is it better to use a sample over a census

• cost effective 

  • conducting a survey on sample saves costs compared to taking a census 

• not feasible 

  • impractical to take census in some situations eg drawing blood 

what is the difference between a population parameter and a sample statistic

population parameter: a numerical fact about the population → obtained via a census

sample statistic: a numerical fact about the sample → obtained from a sample

what is a sample statistic made from

population parameter + bias + random error

what are the 2 statistical inferences

• confidence interval

• hypothesis testing

what is a confidence interval

it is an interval of values computed from sample data that is likely to include the unknown value of a population parameter 

what does a confidence interval consist of

• the interval

• a confidence level

what is the types of confidence intervals

• confidence intervals for population proportion

• confidence intervals for population mean

what is the structure of a confidence interval

sample statistic ± multiplier x standard error 

what is the multiplier

a number based on the confidence level desired and is determined by the standard normal distribution (proportion) / t-distribution (mean

what is the standard error

• an estimate of the standard deviation of the sample statistic

what are the symbols for the sample statistic, multiplier and standard error for a population proportion

sample statistic: p* (sample proportion)

multiplier: z*

standard error: 

√{[p*(1-p*)] / n} where n is the sample size 

“we are 95% confident …” 

what does confident mean in the context of a population proportion

• if many simple random samples obtained + confidence interval constructed for each of them → about 95% of the confidence intervals constructed would contain population parameter 

what are the symbols for the sample stat, multiplier and standard error for a population mean

sample statistic: x̄ 

multiplier: t* (has to be the appropriate value from the student’s t-distribution) 

standard error: s/√n (s: sample standard deviation, n: sample size)

“we are 95% confident …” 

what does confident mean in the context of a population mean

• if many simple random samples obtained + confidence interval constructed for each of them → about 95% of the confidence intervals constructed would contain population parameter 

why is saying “95% chance that the population mean lies between…” wrong?

chances are in sampling procedure (probability), not the parameter

correct way of saying is 95% confident

what are the properties of a confidence interval

1. for the same sample, a change in the confidence level will lead to a change in the size of the confidence interval 

• 90% confidence level → increase in size of confidence interval than 95% confidence level 

2. same confidence level, same sample stat 

• larger sample → confidence interval smaller 

• smaller sample → confidence interval large