Topic 5- Estimation And Confidence Intervals

Question 1

Estimator properties

Answer 1

Unbiased and efficient

Question 2

Efficiency

Answer 2

Is when less dispersed. Less variance=more efficient.

More clustered is good.

Question 3

How to improve efficiency

Answer 3

Increase sample size, as shown by standard deviation of sample mean formula (topic 4, last flash card)

σ/√n.

As we increase n (sample size), standard deviation falls.

Question 4

Point estimate

Answer 4

The statistic, from sample information that estimates a population parameter.

E.g max temp tomorrow will be 15c.

Question 5

Confidence intervals

Answer 5

A range of values that the population parameter is likely to occur within the range found, at a specified probability.

E.g max temperature will be between 13-17C.

Question 6

CI formula

Answer 6

Point estimate +or- margin of error.

Question 7

1.Confidence interval for a population mean with a population SD (σ) known.

2. Confidence interval for a population mean with a population SD (σ) unknown.

Answer 7

1. Sample mean (X bar) ± Zscore (σ/√N)

(σ/√N is the SD of the sample mean remember!)

2. We use sample standard deviation S instead as a population SD is unknown and T STATISTIC NOT Z SCORE

Question 8

Steps to calculate a confidence interval

Answer 8

Decide on confidence level (level of risk)

Find z-score for confidence level. (Divide confidence level equally to find z score.) (Z=x bar-μ/(σ/√n)

Calculate: sub values in CI equation. (Shown earlier)

Question 9

Note: trade off with size of range and confidence level

Answer 9

Increasing confidence makes range wider so less informative.

E.g 95% confidence (alpha=0.05) = x bar+/- 1.96 x sigma/root n
99% confidence has z score 2.575= x bar +/- 2.575 x sigma/root n

+ or - 2.575 is a wider interval, but higher confidence it will lie in there.

Question 10

What is alpha

Answer 10

Alpha is the probability of making an error

E.g if alpha= 0.05, it is saying 5% of times the population mean will not lie in the interval. 95% confidence rate

Question 11

What happens when sigma is unknown

Answer 11

Use s (sample deviation instead), and USE T STATISTIC not z!

Question 12

T statistic

Answer 12

Sample mean - μ
/
S/√N

Question 13

CI for t distribution

Answer 13

Sample mean ± (T using alpha and n-1) x S/√N

Question 14

T distribution features (4)

Answer 14

Mean=0

Standard deviations differ depending on sample size n.

N-1 =degrees of freedom (d.f)

Flatter, less clustered than standard normal.

Question 15

T distribution table- left has degrees of freedom. E.g sample size of 10, we use 9 degrees of freedom (n-1)

Top has the level of significance.

1.What would the t value be for infinity df with 95%?
2.What would the t value be for infinity df with 99% confidence?

Answer 15

1. 1.96 (like z value when alpha is 0.05/ 95% confidence)
2. 2.576 (like z value with 99% confidence)

Question 16

What happens as DF (degrees of freedom) increase

What happpens if N increases

Answer 16

Distribution gets narrower

Question 17

What is a high medium and low confidence level

Answer 17

High=99%
Med=95%
Low=90%

Question 18

Next part: Confidence intervals for a population proportion

A population proportion e.g 60% of McDonalds revenue is through drive through

Question 19

Notations for population proportion vs sample proportion

Answer 19

Population proportion= PIE SYMBOL
Sample proportion= P

Question 20

Sample proportion formula

Answer 20

P=x/n

X is number of successes, N is number sampled.

Question 21

Confidence interval for a population proportion (P)

Answer 21

CI will be

P± Z x √P(1-P)/N

Question 22

How to work out the required sample size to estimate a population mean

Answer 22

N=(Zσ/E)²

Z is standard normal value corresponding to the confidence level e.g 1.96 for 95%
E is maximum allowable error (range of confidence interval)

Question 23

How to work out the required sample size to estimate a population proportion

Answer 23

N= π(1-π) x (Z/E)²

If we can’t find where PP may be, use 0.5

Question 24

When do we use finite population correction factor

Answer 24

use when sample population is not large

Question 25

FPC factor formula, and how do we apply it to confidence intervals to reduce margin of error?

Answer 25

√N-n/N-1

N is total population
n is sample size

Apply at the end of
X bar ± t x S/√N (CI for unknown sigma, so we used S instead of it!)

Question 26

When to use Z and T scores

Answer 26

Z when σ known
T when unknown

This differs slightly in hypothesis where you still use Z if sample is large but σ is unknown!!