Lecture 18- CI mean

Question 1

In a standard normal distribution curve how many standard deviations is 95% of the data within? What is this value know as?

Answer 1

1.96.

The standard normal critical value

Question 2

For a 95% CI what is the alpha value?

Answer 2

1-0.95= 0.05

Therefore each tail will have an area of 0.025

Question 3

What is a 95% confidence interval actually telling us?

Answer 3

Can be 95% certain that the interval contains the true population mean

In other words if 100 samples were done (on average) 95 of them would contain the true population mean and 5 would miss it.

Question 4

What is the general equation for working out a confidence interval?

Answer 4

estimate for the mean _+ multiplier x standard error of the mean

• Where multiplier is the critical value from the sampling distribution
• Also remember that standard error of a sampling distribution is found by dividing the estimate of the standard deviation by the square root of n

Question 5

For a 99% confidence interval what is the multiplier 1.96 replaced with?

Answer 5

a=1 − 0.99 = 0.01
z(1−α/2)
Use qnorm(0.995)

=2.58

Question 6

What confidence interval is wider: 99% or 95%?

Answer 6

99%

We are saying that only 1 out of 100 samples would not include the true population mean. We have greater confidence.

Question 7

What happens to the precision of our confidence interval as n increases?

Answer 7

The standard error of the mean gets smaller and the confidence interval becomes narrower/ more precise i.e we get a better estimate with a larger n value.

Question 8

What do we use a t distribution for?

Answer 8

• Because we don’t actually know the true standard deviation we must estimate it using a sample
• To do this our critical values must now come from the ‘t’ distribution, not the standard normal (as due to sample to sample variation there is less certainty and our 95% confidence interval must reflect this)
• The t-distribution represents a ‘family’ of curves, and using our sample size n, we must specify a value for the degrees of freedom, ν (Greek symbol ‘nu’), to pick out the correct one.

Question 9

How does a t-distribution compare in look to a normal distribution? What does this mean?

Answer 9

• The t-distribution is like the normal distribution, but with fatter tails.
• These fatter tails result in larger critical values that of the normal distribution and hence wider less precise confidence intervals (reflecting greater variation from estimating standard error)

Question 10

Under a t-distribution what does the multiplier for a 95% confidence interval now look like?

Answer 10

(1-a/2, v)

Question 11

For an interval for a single sample mean what do we use as v?

Answer 11

n-1

Question 12

Under what conditions will the t distribution be the correct sampling distribution?

Answer 12

• The underlying distribution of X is normal, and/or

- The sample size is sufficiently large (Central Limit Theorem holds).

Question 13

How do you find the t multiplier in R?

Answer 13

use qt(p, df)

df= degrees of freedom 
p= a probability

Question 14

A pharmacologist is investigating the length of time that a sedative is
effective. Eight patients are selected at random for a study and the eight
times for which the sedative is effective have mean, x = 8.4 hours and
standard deviation, s = 1.5 hours. From previous studies it is known that
the length of time that the sedative is effective is normally distributed.

Find 95% and 99% confidence intervals for the true mean number of
hours the sedative is effective, µX .

Answer 14

• The 95% confidence interval is: (7.15, 9.65)
• The 99% confidence interval is: (6.54, 10.26)

For working look at slides