Transcript Slides for week 11 lecture 1 - Department of Statistics and Probability

Conditions Required for a Valid LargeSample
Confidence Interval for µ
1. A random sample is selected from the target
population.
2. The sample size n is large (i.e., n ≥ 30). Due to the
Central Limit Theorem, this condition guarantees
that the sampling distribution of x is approximately
normal. Also, for large n, s will be a good estimator
of .
Thinking Challenge
• We have a random sample of customer order totals
with an average of $78.25 and a population standard
deviation of $22.5.
• A) Calculate a 90% confidence interval for the mean
given a sample size of 40 orders.
• B) Calculate a 90% confidence interval for the mean
given a sample size of 75 orders.
• C) Explain the difference in the 90% confidence
intervals calculated in A and B.
• D)Calculate the minimum sample size needed to
identify a 90% confidence interval for the mean
assuming a $5 margin of error.
6.3
Confidence Interval for a Population
Mean:
Student’s t-Statistic
Small sample size problem for
inference about 
• The use of a small sample in making inference
about  presents two problems when we
attempt to use the standard normal z as a test
statistic.
Problem 1
• The shape of the sampling distribution of the sample
mean now depends on the shape of the population
sampled.
• We can no longer assume that the sampling
distribution of sample mean is approximately
normal because the central limit theorem ensures
normality only for samples that are sufficiently large.
Solution to Problem 1
• We know that if our sample comes from a population
with normal distribution the sampling distribution of
sample mean will be normal regardless of the sample
size.
Problem 2
• The population standard deviation  is almost
always unknown. For small samples the sample
standard deviaiton s provides poor approximation
for .
Solution to Problem 2
(Small Sample with  known)
Use the standard normal statistic
z
xµ
x
xµ

 n
Solution to Problem 2
(Small Sample with  Unknown)
Instead of using the standard normal statistic
z
use the t–statistic
xµ
x
xµ

 n
xµ
t
s n
in which the sample standard deviation, s, replaces the
population standard deviation, .
Student’s t-Statistic
The t-statistic has a sampling distribution very much like
that of the z-statistic: mound-shaped, symmetric, with
mean 0.
The primary difference
between the sampling
distributions of t and z
is that the t-statistic is
more variable than the
z-statistic.
Degrees of Freedom
The actual amount of variability in the sampling
distribution of t depends on the sample size n. A
convenient way of expressing this dependence is to say
that the t-statistic has (n – 1) degrees of freedom (df).
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’ Tails
t (df = 13)
t (df = 5)
0
The smaller the degrees of freedom for t-statistic, the more variable
will be its sampling distribution.
z
t
• We have a random sample of 15 cars of the same model. Assume
that the gas milage for the population is normally distributed with a
standard deviaition of 5.2 miles per galon.
• A) Identify the bounds for a 90% confidence interval for the mean
given a sample mean of 22.8 miles per gallon.
• B) The car manufacturer of this particular model claims that the
average gas milage is 26 miles per gallon. Discuss the validity of this
claim using the 90% confidence interval calculated in A.
• C) Let a and b represent the lower and upper boundaries of 90%
confidence intervl for the mean of the population. Is it correct to
conclude that tere is a 90% probability that true population mean
lies between a and b?
Thinking Challenge
• In 1882 Michelson measured the speed of
light. His values in km/sec and 299,000
substracted from them. He reported the
results of 23 trials with a mean of 756.22 and
a standard deviaition of 107.12.
• Find a 95% confidence interval for the true
spped of light from these statistics.
• Interpret your result.