Chapter 6 - Confidence Intervals (normality based)

Question 1

Confidence interval

Answer 1

A range of plausible or believable values for an unknown parameter value.

Question 2

We need confidence intervals because…

Answer 2

(point) estimates are almost always wrong. Estimates are just estimates.

Question 3

95% Confidence Interval

Answer 3

A type of interval which contains the true value of a parameter for 95% of samples repeatedly taken in the long run.

Question 4

Parameter

Answer 4

A numerical characteristic of a population or distribution, e.g, a population mean

Question 5

Estimate

Answer 5

A known quantity calculated from the (sample) data to estimate an unknown parameter, e.g., a sample mean is used to estimate the unknown population mean.

Question 6

Standard Error of the sample mean

Answer 6

Measures, roughly, the average distance between a sample mean, x, and the population mean, u

Question 7

For a given sample size, increasing the confidence level:

Answer 7

• Increases the value of the t-multiplier, t
• Increases the width of the confidence interval
• Makes our estimate statement less precise

Question 8

For a given confidence interval, increasing the sample size,n:

Answer 8

• Decreases the value of the standard error of the estimate and in the case of means decreases the t-multiplier
• Decreases the width of the confidence interval
• Makes our estimate statement more precise

Question 9

For a given sample size:

Answer 9

The greater the confidence level, the wider the confidence level.

Question 10

For a given level of confidence:

Answer 10

The bigger the sample size, n, the narrower the confidence interval.

Question 11

Example of sampling situation (a): Proportions from two independent samples.

Answer 11

Random samples of 1000 Australians and 1000 New Zealanders were asked:
“Do you think New Zealand and Australia should become one country?”
36% of the Australians answered ‘Yes’ and 26% of the New Zealanders answered ‘Yes’.
Compare the proportion of all New Zealanders who would have answered ‘yes’ with the proportion of Australians who would have answered ‘yes’.

Question 12

Example of sampling situation (b): One sample size n, several response categories.

Answer 12

A random sample of 1000 New Zealanders were asked:
“Who are you going to vote for in the next election - National, Labour, Green Party, NZ First, Maori Party, or some other party?”
The results were 48%, 32%, 12%, 3%, 2%, and 3% respectively.
Compare the percentage who at the time of the survey intended to vote for National with the percentage who intended to vote for the Maori Party.

Question 13

Example of sampling situation (c): One sample size n, many “Yes/No” items.

Answer 13

A survey of 403 shoppers were asked:
- Do you find TV cameras irritating (Y/N)
- Do you find uniformed guards irritating (Y/N)
- Do you find two way mirrors irritating (Y/N)
24%, 27%, and 29% respectively said ‘Yes’ to each question.
Compare the percentage who find uniformed guards irritating with the percentage who find two-way mirrors irritating.

Question 14

Bootstrap confidence interval

Answer 14

A confidence interval formed using bootstrap resamples taken from the data.

Question 15

Confidence Level

Answer 15

A specified percentage success rate for a method that produces a confidence interval, meaning that the method has this rate for the percentage of times such intervals contain the true value of the population parameter in the long run.

Question 16

Margin of error

Answer 16

An estimate of the likely size of the sampling error in an estimate of a population parameter (“likely” in the sense that the sampling error is very unlikely to be larger than the calculated margin of error).

Question 17

Point Estimate

Answer 17

A single number that estimates the value of a parameter. For example, the value of a sample mean, x, when estimating the value of a population mean, u.