Statistic Ib 7.1
Introduction
Address questions such as,
o How many hours per month, on average, does a college
student spend watching streaming videos on a mobile
phone?
o Do male and female students differ in terms of ‘social
insight,’ the ability to appraise other people?
7.1 Inference for the mean of a population
The sample standard deviation s is used to estimate the
population standard deviation μ
t distributions
Standard error - when the standard deviation of a statistic is
estimated from the data
s
The standard error of a sample mean is SE x́ =
√N
x́−μ
z=
σ /√n
o standard Normal distribution N(0,1)
s/ √ N does not have a Normal distribution; it has a t
distribution
x́−μ
the one-sample t statistic t= has the t distribution
s /√N
with n – 1 degrees of freedom
t(k) stands for the t distribution with k degrees of freedom
the t distributions were discovered in 1908 by William S.
Gosset
the density curve of t(k) distributions are similar in shape to
the standard Normal curve
o they are symmetric about 0; bell shaped
the t distributions have more probability in the tails and less
in the center
o this great spread is due to the extra variability caused
by substituting the random variable s for the fixed
parameter σ
o as the degrees of freedom k increases, the t(k) density
curve gets closer to the N(0,1) curve
table D gives critical values t* for the t distributions
The one-sampled t confidence interval
s
a level C confidence interval for μ is x́ ± t* where t*
√N
is the value fro the t(N-1) density curve with the area C
between –t* and t*
s
margin of error t*
√N
, This interval is exact when the population distribution is
Normal and is approximately correct for large n in other cases
The one-sample t test
H 0 : μ=μ0
x́−μ0
t=
s/√N
In terms of a random variable T having the t(n-1) distribution,
the P-value for a test of H 0 against (p.404)
It is wrong to examine the data first and then decide to do an
one-sided test in the direction indicated by the data
Matched pairs t procedures
In a matched pairs study, subjects are matched in pairs and
their outcomes are compared within each matched pair
The idea is that matched pairs are more similar than
unmatched subjects
Matched pairs are also common when randomization is not
possible
In most circumstances we cannot be absolutely certain about
the direction and the safest strategy is to use the two-sided
alternative
1. A matched pairs analysis is called for when subjects are
matched in pairs or there are two measurements or
observations on each individual and we want to examine the
difference
2. For each pair or individual, use the difference b/w two
measurements as the data for your analysis
3. Use the one-simple confidence interval and significance-
testing procedures
Robustness of the t procedures
The results of one-sample t procedures are exactly correct
only when the population is Normal – real populations are
never exactly Normal
A statistical inference procedure is called robust if the required
probability calculations are insensitive to violations of the
assumptions made
In general, we should be very cautious about discarding
suspected outliers, particularly when they make up a
substantial proportion of the data
Fortunately, the t procedures are quite robust against non-
Normality of the population except in the case of outliers or
strong skewness
Introduction
Address questions such as,
o How many hours per month, on average, does a college
student spend watching streaming videos on a mobile
phone?
o Do male and female students differ in terms of ‘social
insight,’ the ability to appraise other people?
7.1 Inference for the mean of a population
The sample standard deviation s is used to estimate the
population standard deviation μ
t distributions
Standard error - when the standard deviation of a statistic is
estimated from the data
s
The standard error of a sample mean is SE x́ =
√N
x́−μ
z=
σ /√n
o standard Normal distribution N(0,1)
s/ √ N does not have a Normal distribution; it has a t
distribution
x́−μ
the one-sample t statistic t= has the t distribution
s /√N
with n – 1 degrees of freedom
t(k) stands for the t distribution with k degrees of freedom
the t distributions were discovered in 1908 by William S.
Gosset
the density curve of t(k) distributions are similar in shape to
the standard Normal curve
o they are symmetric about 0; bell shaped
the t distributions have more probability in the tails and less
in the center
o this great spread is due to the extra variability caused
by substituting the random variable s for the fixed
parameter σ
o as the degrees of freedom k increases, the t(k) density
curve gets closer to the N(0,1) curve
table D gives critical values t* for the t distributions
The one-sampled t confidence interval
s
a level C confidence interval for μ is x́ ± t* where t*
√N
is the value fro the t(N-1) density curve with the area C
between –t* and t*
s
margin of error t*
√N
, This interval is exact when the population distribution is
Normal and is approximately correct for large n in other cases
The one-sample t test
H 0 : μ=μ0
x́−μ0
t=
s/√N
In terms of a random variable T having the t(n-1) distribution,
the P-value for a test of H 0 against (p.404)
It is wrong to examine the data first and then decide to do an
one-sided test in the direction indicated by the data
Matched pairs t procedures
In a matched pairs study, subjects are matched in pairs and
their outcomes are compared within each matched pair
The idea is that matched pairs are more similar than
unmatched subjects
Matched pairs are also common when randomization is not
possible
In most circumstances we cannot be absolutely certain about
the direction and the safest strategy is to use the two-sided
alternative
1. A matched pairs analysis is called for when subjects are
matched in pairs or there are two measurements or
observations on each individual and we want to examine the
difference
2. For each pair or individual, use the difference b/w two
measurements as the data for your analysis
3. Use the one-simple confidence interval and significance-
testing procedures
Robustness of the t procedures
The results of one-sample t procedures are exactly correct
only when the population is Normal – real populations are
never exactly Normal
A statistical inference procedure is called robust if the required
probability calculations are insensitive to violations of the
assumptions made
In general, we should be very cautious about discarding
suspected outliers, particularly when they make up a
substantial proportion of the data
Fortunately, the t procedures are quite robust against non-
Normality of the population except in the case of outliers or
strong skewness
