• COMM5000 Data Literacy for Business
Inference on the
population - Interval
estimation
⚫ Introduction to Probability
⚫ Probability Concepts
⚫ Probability Rules
⚫ Binomial Distribution
⚫ Poisson Distribution
⚫ Normal Distribution
Agenda
• A sampling distribution is a distribution of all of the
possible values of a sample statistic for a given size
sample selected from a population.
• We make inferences about the characteristics of the
population based on a random sample.
• Specifically, we estimate a population parameter based
on a sample statistic
Sampling Distributions
Standard Error of the Mean
• Different samples of the same size from the same
population will yield different sample means
• A measure of the variability in the mean from sample to
sample is given by the Standard Error of the Mean:
Note that the standard error of the mean decreases as the
sample size increases
Sampling Distributions
n
σ
σ
X
=
Sample Mean Sampling Distribution:
If the Population is Normal
• If a population is normal with mean μ and standard deviation σ,
the sampling distribution of ത is also normally distributed with
an expected value of
ത =
and a standard error of
ത =
Where standard error of the mean is the standard deviation of the
sampling distribution and is a measure of the variability in the
mean from sample to sample
Sampling Distributions
• Z-value for the sampling distribution of ത:
Sampling Distributions
n
σ
μ)X(
σ
)μX(
Z
X
X −=
−
=
where: ത = sample mean
μ = population mean
= population standard deviation
n = sample size
• We can apply the Central Limit Theorem:
• Even if the population is not normal,
• sample means from the population will be
approximately normal as long as the sample size is
large enough.
• But how large is large Enough?
• For most distributions, n > 30 will give a sampling
distribution that is nearly normal
• For normal population distributions, the sampling
distribution of the mean is always normally distributed
Sampling Distributions
Suppose a population has mean μ = 8 and standard deviation σ
= 3. Suppose a random sample of size n = 36 is selected. What
is the probability that the sample mean is between 7.8 and
8.2?
Even if the population is not normally distributed, the central
limit theorem can be used (n > 30) so the sampling distribution
of ത is approximately normal with an expected mean E( ത) = 8
and standard deviation:
Example
0.5
36
3
n
σ
σx ===
Example
0.3108 0.3446 - 0.65540.4)ZP(-0.4
36
3
8-8.2
n
σ
μ- X
36
3
8-7.8
P 8.2) X P(7.8
===










=
Suppose a population has mean μ = 8 and standard
deviation σ = 3. Suppose a random sample of size n = 36 is
selected. What is the probability that the sample mean is
between 7.8 and 8.2?
Population Proportions
P = the proportion of the population having some characteristic e.g. proportion of
people in the population who are left handed
• Sample proportion ത =
provides an estimate of p:
• By the CLT, the sampling distribution of ത is approximately normal when ≥ 5
and (1 − ) ≥ 5.
• The standard error of the proportion ത =
(1−)

• Z formula becomes Z =
ത −
Sep
=
ത −
(1 − )
n
Example
If the true proportion of voters who support Proposition A is P = 0.4, what is the
probability that a sample of size 200 yields a sample proportion between 0.40
and 0.45?
– i.e. P = 0.4 and n = 200, what is Prob (0.40 ≤ ത ≤ 0.45)
Sep =
(1 − )
n
=
0.4(1 − 0.4)
200
= 0.03464
ത(0.40 ≤ p ≤ 0.45) = P
0.40 − 0.40
0.03464
≤ Z ≤
0.45 − 0.40
0.03464
= P(0 ≤ Z ≤ 1.44)
P(0 ≤ Z ≤ 1.44) = 0.9251 – 0.5000 = 0.4251 or approx. 42.51%
Interval Estimates
A point estimate is a single number
• a confidence interval (interval estimate) provides more
information about a population characteristic than does a
point estimate
• it is our point estimate plus or minus a margin of error
Point Estimate
Lower Limit
Width of
confidence interval
Upper Limit
Interval Estimates
The general formula for all confidence intervals is:
Point Estimate ± (Margin of Error)
Where:
• Point Estimate is the sample statistic estimating the population
parameter of interest
• Margin of error is Critical Value * Standard error:
• Critical Value is a table value based on the sampling distribution of
the point estimate and the desired confidence level
• Critical value is either a Z or t statistic
• Standard Error is the standard deviation of the point estimate
Confidence Intervals
Population Mean
σ Unknown
Confidence intervals
Population Proportionσ Known
X ± Z/2,
σ
n ҧ ± Τ 2,
ҧ ± Τ 2
ҧ(1 − ҧ)
Confidence Interval Examples
When we Know
• A random sample of 11 shares were drawn from a portfolio with normal distribution.
Mean return of the sampled shares is 2.20% p.a. We know from historical data that
the population standard deviation is 0.35 %.
• Determine a 95% confidence interval for the true mean share return of the population.
X ± Z/2 ,
σ
n
= 2.20 ± 1.96 (0.35/ 11)
= 2.20 ± 0.2068
1.9932 ≤ ≤ 2.4068
Confidence Interval Examples
When we Don’t Know
A random sample of n = 25 has X = 50 and S = 8. Form a 95%
confidence interval for μ
46.698 ≤ μ ≤ 53.302
± /2,
S
n
= 50 ± (2.0639)
8
25
Confidence Interval Examples
When we Don’t Know ( )
Use the data provided in Excel file Finance.xlsx. The data compare the
performance of the stocks of two Internet companies, Amazon (AMZN) and
Microsoft Corporation (MSFT). The data show the daily closing prices of the two
stocks for the last 100 days.
- Open the Data file
- Conduct the variable transformation by calculating the daily returns for both
companies based on closing prices
- Estimate the 95% Confidence Interval for daily stock returns of both companies
using:
a) Excel T.INV() formula
b) Excel Analysis Toolpack
Confidence Interval Examples
a) Excel T.INV() formula
b) Excel Analysis Toolpack
Q&A