Confidence intervals 2: Theory in practice

Question 1

For 95% of the repeats m would be within …

Answer 1

1) 1.96 standard errors of the mean
2) Between
(mean - (1.96 x SDM SD) to µ + (1.96 x SDM SD) )

Question 2

For a sample (N = 15) drawn at random from a normal population

N(µ = ?, 8.9) with known SD 8.9 and sample mean m = 163.2

The SDM has parameters
N(µ = ?, 8.9/√15) = N(µ = ?, 2.3)
Sample mean = 163.2

Calculate the 95% CI

Answer 2

mean - (1.96 x SDM SD)
163.2 - (1.96 x 2.3) = 158.69

mean + (1.96 x SDM SD)
163.2 + (1.96 x 2.3) = 167.71

95% CI. = (158.69, 167.71)

Question 3

What do we calculate 95% CI if the population SD is unknown?

Answer 3

Using estimated standard error instead of standard error

sample mean + (c x e.s.e.)
sample mean - (c x e.s.e.)

Question 4

A research team on earth has conducted a large scale observation of height (in cm) of all visiting aliens from the planet Ziltoidia 10. The population standard deviation of heights is recorded as 3.2cm but, sadly, the population mean and all the other data records have been lost after one of the research team spilled a bad cup of coffee all over the external hard drive containing the data.

You get a random sample of N=16 Ziltoidians from the visiting population and measure their heights. The sample mean height in cm is 17.4cm.

What is the 95% confidence interval for the population mean centred on the sample mean?

a.	(16.6, 18.2)

b.	(17.008, 17.792)

c.	(11.128, 23.672)

d.	(15.832, 18.968)

Answer 4

95% CI =
mean - (1.96 x s.e. or SDM SD)
mean + (1.96 x s.e. or SDM SD)

s.e. = pop SD / SQRT (sample size)
s.e. = 3.2 / SQRT (16)
s.e. = 0.80

17.4 - (1.96 x 0.80) = 15.832
17.4 + (1.96 x 0.80) = 18.968

Answer = D

Question 5

A research team on earth has conducted a large scale observation of height (in cm) of all visiting aliens from the planet Ziltoidia 10. The population standard deviation of heights is recorded as 3.2cm but, sadly, the population mean and all the other data records have been lost after one of the research team spilled a bad cup of coffee all over the external hard drive containing the data.

You get a random sample of N=25 Ziltoidians from the visiting population and measure their heights. The sample mean height in cm is 17.4cm.

What is the 95% confidence interval for the population mean centred on the sample mean?

a.	(16.1456, 18.6544)

b.	(11.128, 23.672)

c.	(17.1491, 17.6509)

d.	(16.76, 18.04)

Answer 5

95% CI =
mean - (1.96 x s.e.)
mean + (1.96 x s.e.)

s.e. = pop SD / SQRT (sample size)
s.e. = 3.2 / SQRT (25)
s.e. = 0.64

17.4 - (1.96 x 0.64) = 16.1456
17.4 + (1.96 x 0.64) = 18.6544

Answer = A

Question 6

A research team on earth has conducted a large scale observation of IQ of all visiting aliens from the planet Ziltoidia 10. The population standard deviation of IQ is recorded as 12.65 but, sadly, the population mean and all the other data records have been lost after an accident caused by human error.

You get a random sample of N=36 Ziltoidians from the visiting population and measure their IQ. The sample mean IQ is 125.9.

What is the 95% confidence interval for the population mean for IQ centred on the sample mean (give the answer to 4 d.p.)?

a.	(125.2113, 126.5887)

b.	(123.7917, 128.0083)

c.	(121.7677, 130.0323)

d.	(101.1060, 150.6940)

Answer 6

95% CI =
mean - (1.96 x s.e. or SDM SD)
mean + (1.96 x s.e. or SDM SD)

s.e. = pop SD / SQRT (sample size)
s.e. = 12.65 / SQRT (36)
s.e. = 2.11

125.9 - (1.96 x 2.11) = 121.7677
125.9 + (1.96 x 2.11) = 130.0323

Answer = C