Confidence Intervals For the Mean

Question 1

what are confidence intervals for?

Answer 1

statistical inference draws conclusions about a population or process based in sample data and gives a statement in probability terms of how much confidence we can place in conclusions.

Question 2

confidence interval equation

Answer 2

sample value +/- (multiplier x standard error)

Question 3

when the sigma is known for 95%

Answer 3

on average 95% of 95% confidence intervals contain the mean

mew is x bar +/- 1.96 sigma/ square root (n)

P(Z<1.96) is 0.975
P(Z>1.96) is 0.025
P(between -1.96 and 1.96) is 0.95
Z0.025 = 1.96 (2.5 is above and 2.5 is below)

Question 4

what are aspects of confidence intervals

Answer 4

interval and confidence level

Question 5

90% confidence interval when sigma is known

Answer 5

mew is x bar +/- 1.645 sigma/ square root (n)

P(Z<1.645) is 0.95
P(Z>1.645) is 0.05
P(between -1.645 and 1.645) is 0.90
Z0.05 = 1.645 (5% is above and 5% is below)

Question 6

99% CI when sigma is known

Answer 6

mew is x bar +/- 2.5758 sigma/ square root (n)

Z0.01 = 2.5758 (0.5% is above and 0.5% is below)

Question 7

general CI rule when sigma is known

Answer 7

100(1-a)% CI for mew is x bar +/- Za/2 . sigma/square root (n)

Question 8

confidence interval using central limit theorum

Answer 8

The others assume population is normal. If not normal but n is large enough:

X bar is approx. normal so (x bar - mew)/(sigma/ square root (n)) - this is a test statistic and can then use the general CI

Question 9

choosing sample size

Answer 9

want to know how many samples to take to achieve a certain accuracy (or error)

must know:

1. what CI we want
2. how accurate they want their CI to be eg. d=0.8
3. population SD

n = ((Za/2 . sigma)/d) squared

Question 10

confidence interval when sigma is unknown

Answer 10

without sigma, we cannot use normal distribution
use t distributions (longer tails, give larger confidence intervals making it harder to reject the null)
must assume the pop is normal
must estimate sample standard deviation (s)
degrees of freedom (df or r) are n-1

Question 11

sample size (n) is taken from a normal distribution and gibes x bar, s then the 95% CI is

Answer 11

x bar +/- ta/2, n-1 . s/squareroot (n)

Question 12

general CI when sigma is unknown

Answer 12

x bar +/- ta/2 . s/square root (n)