Confidence Intervals

Question 1

What is a point estimate?

Answer 1

is a single value estimate of a population parameter e.g. a sample mean

Question 2

What is an interval estimate?

Answer 2

a range of possible values of a population parameter: e.g. a confidence interval

Question 3

What does a confidence interval (CI) describe?

Answer 3

A confidence interval (CI) describes an interval (i.e. a range) of values for our population parameter, together with a specified level of confidence that the parameter is in that range

Question 4

When given the question: • How big should the range be for us to have 95% confidence that it contains pop mean? what should you consider to work this out and what does this mean (remembering that we are interested in a sample mean and its relation to the population mean) (we have the sample statistics and the population standard deviation but not the population mean)

Answer 4

• sensible to consider the sampling distribution of the mean
• the SDM is normal so a z-transformation can take place
• the area under the SND between z= -1.96 and z = 1.96 is 0.95 (95%)
• therefore 95% of our z-scores are within 1.96 standard deviations of the population mean
• and if we repeated our data collection many times then 95% of sample means would be within 1.96 standard errors of the population mean

Question 5

What is the equation that says that 95% of sample means would be within 1.96 standard errors of the population mean?

Answer 5

u - (1.96 x standard error) to u+ (1.96 x standard error)

Question 6

If 95% of our sample means would be within 1.96 standard errors of our population mean what else does this mean?

Answer 6

• That 95% of our population means would be within 1.96 standard errors of our sample mean
• therefore 95% of repeats would be within:
• m – (1.96 x standard error) to m +(1.96 x standard error)

Question 7

What is the basic principles of confidence interval theory?

Answer 7

• For a sample size of size N drawn from a random from a normal population N(u, standard deviation) with known s.d.:
• The 95% CI for the population mean is centred on the sample mean mm and goes from m- (1.96 x standard error) to m +(1.96 x standard error)
• C.I lower = -1.96 standard errors
• C.I upper = + 1.96 s.e.s
• A 95% confidence level means that if we repeated our sampling many times and worked out a new CI each time that was centred on our new sample mean we would expect the population mean to be in that interval on 95% of repeats

Question 8

Calculate the confidence intervals at 95% for a sample size of 15 with population parameters: N(?,8.9) and sample statistics: N(163.2,14.3)

Answer 8

• we can work out our standard error: 8.9/square root 15 = 2.30
• for 95% repeats u would be within 1.96 standard errors of m:
• m -(1.96 x s.e) to m +(1.96 x s.e.)
• 163.2 - (1.96 x 2.3) to 163.2 + (1.96 x 2.3)
• C.I. lower = 163.2 - (1.96 x 2.3) = 158.69
• C.I. upper = 163.2 + (1.96 x 2.3) = 167.71
• 95% C.I = (138.69, 167.71)