Mid-Term Review

Final Review

Statistical for Business (1)(2)

Mid-Term Review

Remarks

This is a close-book examination. Calculation can be carried out in a calculator.

Other computational tools, such as mobile phone, are prohibited.

There are 8 questions for a total of 100 points. A formula bank is attached (from Chap. 7).

Time: 7 Jun 2010 (Mon) 9:30am-11:30pm

Mid-Term Review

Chapter 4A A random variablerandom variable is a variable that assumes numerical is a variable that assumes numerical values that are determined by the outcome of an experimentvalues that are determined by the outcome of an experiment.

The probability distribution of a discrete random variable is a table, graph, or formula that gives the probability associated with each possible value that the variable can assume

Properties

1. For any value x of the random variable, p(x) 0

2. The probabilities of all the events in the sample space must sum to 1, that is,

Mid-Term Review

The Binomial DistributionThe Binomial Experiment:1. Experiment consists of n identical trials2. Each trial results in either “success” or “failure”3. Probability of success, p, is constant from trial to trial4. Trials are independent

If If xx is the total number of successes in is the total number of successes in nn trials of a binomial trials of a binomial experiment, then experiment, then xx is a is a binomial random variablebinomial random variable

x-nxqp

x-nx

n =xp

!!

!

Number of ways to get x successes and (n–x) failures in n trials

The chance of getting x successes and (n–x)

failures in a particular arrangement

Mid-Term Review

If x is a binomial random variable with parameters n and p (so q = 1 – p), then

npq

npq

np

X

X

X

deviation standard

variance

mean

2

Mid-Term Review

The Poisson DistributionIf x = the number of occurrences in a specified interval, then x is a Poisson random variable

!x

exp

x

where x can take any of the values x = 0, 1, 2, 3, …

and e = 2.71828… (e is the base of the natural logs)

X

X

X

deviation standard

variance

mean

2

Mid-Term Review

Chapter 5Recall: A continuous random variable may assume any numerical value in one or more intervals

Use a continuous probability distribution to assign probabilities to intervals of values

Properties of f(Properties of f(xx):): f( f(xx) is a continuous function such ) is a continuous function such thatthat

1.1. f(f(xx) ) 0 for all 0 for all xx2.2. The total area under the curve of f(The total area under the curve of f(xx) is equal to 1) is equal to 1

Essential point: An area under a continuous probability Essential point: An area under a continuous probability distribution is a probabilitydistribution is a probability

Mid-Term Review

The Uniform Distribution

dx c cd=x

otherwise0

for1

f

The mean X and standard deviation X of a uniform random variable x are

12

2

cd

dc

X

X

Mid-Term Review

The Normal Distribution

2

1=)f(

2

2

1

eπσ

x

x

If x is normally distributed with mean and standard deviation , then the random variable z

x

z

is normally distributed with mean 0 and standard deviation is normally distributed with mean 0 and standard deviation 1; this normal is called the 1; this normal is called the standard normal distribution.standard normal distribution.

Mid-Term Review

Chapter 6: Distribution of Sample Mean (Central Limit Theorem)(Central Limit Theorem)

x

Sample mean is approximately normally distributed if:

Sample size is larger than 30 (without assuming the population also has a normal distribution);

Sample size is less than 30, and the population also has a normal distribution.

– Sample mean μ Population mean – Mean of sample mean σ– Population standard deviation

– Standard deviation of sample mean n- sample size

xXX

σ

When sample size is large:

2

( , )x Nn

μμx n

σσx approximately~

Mid-Term Review

Chapter 7 z-Based Confidence Intervals for aMean with Known• If a population has standard deviation (known),

• and if the population is normal or if sample size is large (n 30), then …

• … a )100% confidence interval for is

nzx,

nzx

nzx 222

Mid-Term Review

t-Based Confidence Intervals for aMean: Unknown

If the sampled population is normally distributed with mean , then a )100% confidence interval for is

n

stx 2

Sample standard deviation

t/2 is the t point giving a right-hand tail area of /2 under the t curve having n – 1 degrees of freedom

Mid-Term Review

z-Based Confidence Intervals for a Population Proportion

If the sample size n is large*, then a )100% confidence interval for p is

n

p̂p̂zp̂

12

* Here n should be considered large if both

5ˆ15ˆ pnandpn

Mid-Term Review

Sample Size DeterminationSample Size Determination

2

2

E

zn

Letting E denote the desired margin of error, so that is within E units of , with 100(1-)% confidence.

x

If If σ is unknown and is estimated from is unknown and is estimated from ss2

2

E

stn

so that is within E units of , with 100(1-)% confidence. The number of degrees of freedom for the t/2 point is the size of the preliminary sample minus 1

x

If If σ is known is known

Mid-Term Review

A sample size

2

21

E

zppn

will yield an estimate , precisely within E units of p, with 100(1-)% confidence

Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p

p̂

Sample Size Determination Sample Size Determination for p

Mid-Term Review

Chapter 8: Six Step Model for Hypothesis Tests

Step 1: State the null and alternate hypothesis

Step 2: Select a level of significance

Step 3: Identify a test statistics

Step 4: Determine the critical value and rejection region

Step 5: Take a sample and compute the value of the test statistics

Step 6: Make the statistical decision:

Do not reject null Reject null and accept alternate

Mid-Term Review

Hypotheses H0: H0: H1:H1:

Test Statistic Test Statistic KnownKnown known known

Rejection Rule Reject Rejection Rule Reject HH0 0 if if zz > > zzReject Reject HH0 0 if if zz < - < -zz

One-Tailed Tests about a Population Mean:

z xn

0

/z x

n

0

/ nxz

/0

n

xz/

0

00 zα=1.65 zα=1.65

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

(Critical value)(Critical value)z

0.05 Level of Significance:αA right-TailedTestH1: μ > μ0

00 -zα=-1.65 -zα=-1.65

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

(Critical value)(Critical value)z

A left-Tailed TestH1: μ < μ0

Mid-Term Review

Hypotheses: H0: H1: Test Statistic Known

Rejection Rule Reject H0 if |z| > z

Two-Tailed Tests about a Population Mean

z xn

0

/z x

n

0

/

00 1.96 1.96

Reject H0Reject H0Do Not Reject H0Do Not Reject H0

zz

Reject H0Reject H0

-1.96 -1.96

0.05 Level of Significance:α

Mid-Term Review

Tests about a Population Mean:Tests about a Population Mean:σ Unknown

Test Statistic:Test Statistic: σ Unknown

This test statistic has a This test statistic has a tt distribution with distribution with nn - 1 degrees - 1 degrees of freedom.of freedom.

txs n

0

/t

xs n

0

/

Rejection Rule Rejection Rule

HH00: : Reject Reject HH0 0 if if tt > > tt HH00: : Reject Reject HH0 0 if if tt < - < -tt HH00: : Reject Reject HH0 0 if |if |tt| | > > tt

Mid-Term Review

Test about a Population ProportionTest about a Population Proportion Hypotheses (where p0 is the hypothesized value of the

population proportion).

H0: p > p0 H0: p < p0 H0: p = p0

H1: p < p0 H1: p > p0 H1: p ≠ p0

Left-tailedLeft-tailed Right-tailedRight-tailed Two-tailedTwo-tailedTest StatisticTest Statistic

p

ppz

ˆ

0ˆ

p

ppz

ˆ

0ˆ

n

ppp

)1( 00ˆ

n

ppp

)1( 00ˆ

Where:

Rejection RuleRejection Rule

HH00: : pppp Reject Reject HH0 0 if z > zif z > z HH00: : pppp Reject Reject HH0 0 if z < -zif z < -z HH00: : pppp Reject Reject HH0 0 if |z| if |z| > > zz

Mid-Term Review

Chapter 9: Two-Sample Tests

Two-Sample Tests

Population Means,

Independent Samples

Means, Related Samples

Population 1 vs. independent Population 2

Same population before vs. after treatment

Examples:

Mid-Term Review

Population means, independent samples

σ1 and σ2 known

2

22

1

21

2121

nσ

nσ

μμXXZ

The test statistic for μ1 – μ2 is:

σ1 and σ2 unknown, assumed equal

σ1 and σ2 unknown, not assumed equal

21

2p

2121

n1

n1

S

μμXXt

Where t has (n1 + n2 – 2) d.f.,

1)n()1(n

S1nS1nS

21

222

2112

p

Mid-Term Review

Test of Related Populations

A paired t test: The test statistic for d is a t statistic, with n-1 d.f.:

Tests Means of 2 Related PopulationsPaired or matched samplesRepeated measures (before/after) Use difference between paired values

di = X1i - X2i (The ith paired difference is di )

d

d μt

S

n

d

n2

ii 1

d

(d d)S

n 1

Where Sd is: