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### Quantitative Business Methods for Decision Making

Estimation and Testing of Hypotheses

Lecture Outlines

Estimation

- Confidence interval for estimating means
- Confidence interval for predicting a new observation
- Confidence interval for estimating proportions

403.6

Lecture Outlines (con’t)

Hypothesis Testing

- Null and alternative hypotheses
- Decision rules (Tests) and their level of significance
- Type I and Type II errors
- Tests of hypotheses for comparing means
- Tests of hypotheses for comparing proportions

403.6

Estimating a Population Mean

- Population mean is estimated by , the sample mean
- Standard error of , i.e.

will decrease as n gets large.

403.6

Confidence Interval for Estimating if is known

With a 95% degree of confidence is estimated within ( )

written as Or more accurately

by

403.6

Confidence Interval for if is not known

Use instead of ,

remember , and

“t” is 95th% percentile of the t

distribution with (n-1) degrees of freedom.

403.6

An Illustration

Suppose n= 26. Then degrees of freedom

(d.f.) = n-1 = 25.

A two-sided degree of C.I. is computed

by

But, for a one-sided 95% C.I. , t = 1.711 instead

of 2.064

403.6

Assumptions and Sample Size forEstimation of the mean

The population should be normally (at least

close to) distributed. If skew, then median is

an appropriate measure of the center than the

mean.

To estimate mean with a specified margin of

error (m.e.), take a random sample of size n

large size.

403.6

Prediction Interval for a New Observation on X

Prediction Interval for a new observation is given by

403.6

Confidence Interval for a Population Proportion

Let denote the proportion of items in a

population having a certain property

An estimate of is the binomial

proportion: , What is ?

For a C.I. for , use

403.6

Confidence Intervals for the Proportion (con’t)

For estimating ,“t” is the percentile of the

t-distribution with (equivalently,

percentile of the standard normal

distribution), and s.e. of p is

403.6

Hypotheses Testing

- The hypothesis testing is a methodology for proving or disproving researcher’s prior

beliefs.

- Statements that express prior beliefs are

framed as alternative hypotheses.

- Complementary statement to an

alternative hypothesis is called null

hypothesis.

403.6

Null and Alternative

Ha: Researcher’s belief that are to be tested (alternate hypothesis)

H0: Complement of Ha (Null hypothesis)

403.6

Statistical Decision

- A decision will be either:
- Reject H0 (Ha is proved)
- or
- Do not reject H0 (Ha is not proved)

403.6

Hypothesis Testing Methodology for the mean

Depending upon what an investigator

believes a priori, an alternative hypothesis

is formulated to be one of the followings:

1.

2.

3.

one-sided

403.6

A Test Statistic

Regardless of what an alternative hypothesis

about the mean is formulated, the decision

rule is defined by a t- statistic:

403.6

Comparing Two Means

The reference number is a specified amount for comparing the

difference between two means. There are two distinct practical

situations resulting in samples on X and Y.

403.6

Paired Sample

- Two variables X and Y are observed for each unit in the sample to measure the same aspect but under two different conditions.
- Thus, for n randomly selected units, a sample of n pairs (X, Y) is observed.
- Compute differences: X1-Y1= d1, X2-Y2= d2, etc. and then mean
- Compute Sd of differences

403.6

Paired Samples (con’t)

- Reject H0 if absolute value of t-statistic is more than
- the desired percentile of the t-distribution.
- Alternatively, find the p value of the t-statistics and
- reject H0 if the p value is less than the desired
- significance level.

403.6

Two Independent (Unpaired) Samples

- Populations of variables X and Y (for

example, males salary X and females

salary Y).

- Take samples independently on X and Y.
- Compute
- Compute pooled standard deviation

403.6

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