Ka-fu Wong © 2003 Chap 8- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.

Ka-fu Wong © 2003 Chap 8- 1

Dr. Ka-fu Wong

ECON1003Analysis of Economic Data


Central Limit Theorem #1

5 balls in the bag:

Draw 50 ball 1000 times with replacement. Compute the sample mean. Plot a relative frequency histogram (empirical probability histogram) of the 1000 sample means.

0 1 2 3 4

The Central Limit Theorem says 1. The empirical histogram looks like a normal density.2. Expected value (mean of the normal distribution) = 2.3. Variance of the sample means = 2/50=0.04.


Confidence Interval #1

Five numbered balls in the bag:

Draw one sample of 50 balls with replacement. Compute the sample mean and sample standard deviation. Suppose the sample mean is 10 and the sample standard deviation is 0.04. Can you tell us the range of possible values the population mean may take, at 95% confidence level?

? ? ? ? ?


Hypothesis testing #1

Five numbered balls in the bag:

Draw one sample of 50 balls with replacement. Compute the sample mean and sample standard deviation. Suppose the sample mean is 10 and the sample standard deviation is 0.04. Do you think the balls in this bag has a mean of 2?

? ? ? ? ?

2

Ka-fu Wong © 2003 Chap 8- 5l

GOALS

1. Define a hypothesis and hypothesis testing.2. Describe the five step hypothesis testing

procedure.3. Distinguish between a one-tailed and a two-

tailed test of hypothesis.4. Conduct a test of hypothesis about a

population mean.5. Conduct a test of hypothesis about a

population proportion.6. Define Type I and Type II errors.7. Compute the probability of a Type II error.

Chapter EightOne-Sample Tests of One-Sample Tests of HypothesisHypothesis


What is a Hypothesis?

A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing.

Examples of hypotheses made about a population parameter are:The mean monthly income for systems

analysts is $3,625.Twenty percent of all customers at

Bovine’s Chop House return for another meal within a month.


What is Hypothesis Testing?

Hypothesis testing is a procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.


Hypothesis Testing

Step 1: state null and alternative hypothesis

Step 2: select a level of significance

Step 3: identify the test statistic

Step 4: formulate a decision rule

Step 5: Take a sample, arrive at a decision

Do not reject null Reject null and accept alternative


Definitions

Null Hypothesis H0: A statement about the value of a population parameter.

Alternative Hypothesis H1: A statement that is accepted if the sample data provide evidence that the null hypothesis is false.

Level of Significance: The probability of rejecting the null hypothesis when it is actually true.


Objectivity in formulating a hypothesis

In court, the defendant is presumed innocent until proven beyond reasonable doubt to be guilty of stated charges.

The “null hypothesis”, i.e. the denial of our theory, is presumed true until we prove beyond reasonable doubt that it is false. “Beyond reasonable doubt” means that the

probability of claiming that our theory is true when it is not (null hypothesis true) is less than an a priori set significance level (usually 5% or 1%).

Is the defendant guilty? Null: the defendant is not guilty. Alternative: the defendant is guilty.


Definitions

Type I Error: conclude the defendant guilty when the defendant did not commit the crime. Level of significance is also the maximum

probability of committing a type I error. We want to limit this Type I Error to some small number.

Type II Error: Conclude the defendant not guilty when the defendant actually committed the crime.

Committed Crime

Yes No

Court Decision(Guilty or

not)

Guilty Correct decision

Type I error

Not Guilty Type II error Correct decision


Definitions

Type I Error: Rejecting the null hypothesis when it is actually true. Level of significance is also the maximum

probability of committing a type I error. We want to limit this Type I Error to some small number.

Type II Error: Accepting the null hypothesis when it is actually false.

State of nature

Null true Null false

Decision based on

the sample statistic

Don’t reject null

Correct decision

Type II error

Reject null Type I error Correct decision


Definitions

Test statistic: A value, determined from sample information, used to determine whether or not to reject the null hypothesis.

Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.


One-Tailed Tests of Significance

A test is one-tailed when the alternate hypothesis, H1 , states a direction, such as:

H1: The mean yearly commissions earned by full-time realtors is more than $35,000. (µ>$35,000)

H1: The mean speed of trucks traveling on I-95 in Georgia is less than 60 miles per hour. (µ<60)

H1: Less than 20 percent of the customers pay cash for their gasoline purchase. ( < .20)


0 1 2 3 4

CriticalValuez=1.65

.95 probability

Sampling Distribution for the Statistic Z for aOne Tailed Test, .05 Level of Significance

Rejection regionReject the null if the test statistic falls into this region.

.05 probability


Two-Tailed Tests of Significance

A test is two-tailed when no direction is specified in the alternate hypothesis H1 , such as:

H1: The mean amount spent by customers at the Wal-Mart in Georgetown is not equal to $25. (µ $25).

H1: The mean price for a gallon of gasoline is not equal to $1.54. (µ $1.54).

Ka-fu Wong © 2003 Chap 8- 17Copyright© 2002 by The McGraw-Hill Companies, Inc. All rights reserved

-4 -3 -2 -1 0 1 2 3 4

CriticalValuez=1.96

.95 probability

Sampling Distribution for the Statistic Z for aTwo Tailed Test, .05 Level of Significance

Rejection region #2

Reject the null if the test statistic falls into these two regions.

Rejection region #1

.025 probability

.025 probability

CriticalValuez=-1.96


Testing for the Population Mean: Large Sample, Population Standard Deviation Known

When testing for the population mean from a large sample and the population standard deviation is known, the test statistic is given by:

n/

X

z


EXAMPLE 1

The processors of Fries’ Catsup indicate on the label that the bottle contains 16 ounces of catsup. The standard deviation of the process is 0.5 ounces. A sample of 36 bottles from last hour’s production revealed a mean weight of 16.12 ounces per bottle. At the .05 significance level is the process out of control?

That is, can we conclude that the mean amount per bottle is different from 16 ounces?


EXAMPLE 1 continued

Step 1: State the null and the alternative hypotheses:

H0: = 16; H1: 16

Step 2: Select the level of significance. In this case we selected the .05 significance level.

Step 3: Identify the test statistic. Because we know the population standard deviation, the test statistic is z.


EXAMPLE 1 continued

Step 4: State the decision rule: Reject H0 if z > 1.96 or z < -1.96

Step 5: Compute the value of the test statistic and arrive at a decision.

44.1365.0

00.1612.16

n

Xz

Do not reject the null hypothesis. We cannot conclude the mean is different from 16 ounces.


p-Value in Hypothesis Testing

A p-Value is the probability, assuming that the null hypothesis is true, of finding a value of the test statistic at least as extreme as the computed value for the test. The “critical probability” for our decision to reject

the null.

If the p-Value is smaller than the significance level, H0 is rejected.

If the p-Value is larger than the significance level, H0 is not rejected.


Computation of the p-Value

One-Tailed Test: p-Value = P{z ≥absolute value of the computed test statistic value}

Two-Tailed Test: p-Value = 2P{z ≥ absolute value of the computed test statistic value}

From EXAMPLE 1, z = 1.44, and because it was a two-tailed test, the p-Value = 2P{z ≥ 1.44} = 2(.5-.4251) = .1498. Because .1498 > .05, do not reject H0.


Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown

Here is unknown, so we estimate it with the sample standard deviation s.

As long as the sample size n 30, z can be approximated with:

ns

Xz

/


EXAMPLE 2

Roder’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400. The level of significance is set at .05. A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38. Should Lisa conclude that the population mean is greater than $400, or is it reasonable to assume that the difference of $7 ($407-$400) is due to chance?


EXAMPLE 2 continued

Step 1: H0: $400, H1: > $400 Step 2: The significance level is .05 Step 3: Because the sample is large we can use the z

distribution as the test statistic. Step 4: H0 is rejected if z>1.65 Step 5: Perform the calculations and make a decision.

42.217238$

400$407$

ns

Xz

H0 is rejected. Lisa can conclude that the mean unpaid balance is greater than $400.


Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown

The test statistic is the t distribution.

The test statistic for the one sample case is given by:

ns

Xt

/


Example 3

The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. At the .05 significance level can Neary conclude that the new machine is faster?


Example 3 continued

Step 1: State the null and the alternate hypothesis. H0: 250; H1: > 250

Step 2: Select the level of significance. It is .05. Step 3: Find a test statistic. It is the t distribution

because the population standard deviation is not known and the sample size is less than 30.


Example 3 continued

Step 4: State the decision rule. There are 10 – 1 = 9 degrees of freedom. The null hypothesis is rejected if t > 1.833.

Step 5: Make a decision and interpret the results.

162.3106

250256

ns

Xt

The null hypothesis is rejected. The mean numberproduced is more than 250 per hour.


Tests Concerning Proportion

A Proportion is the fraction or percentage that indicates the part of the population or sample having a particular trait of interest.

The sample proportion is denoted by p and is found by:

sampled Number

sample the in successes of Numberp


Test Statistic for Testing a Single Population Proportion

n

pz

)1(

The sample proportion is p and is the population proportion.


EXAMPLE 4

In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution. A new solicitation letter that has been drafted is sent to a sample of 200 people and 45 responded with a contribution. At the .05 significance level can it be concluded that the new letter is more effective?


Example 4 continued

Step 1: State the null and the alternate hypothesis. H0: .15 H1: > .15

Step 2: Select the level of significance. It is .05.

Step 3: Find a test statistic. The z distribution is the test statistic.


Example 4 continued

Step 4: State the decision rule. The null hypothesis is rejected if z is greater than 1.65.

Step 5: Make a decision and interpret the results.

97.2

200)15.1(15.

15.20045

)1(

n

pz

The null hypothesis is rejected. More than 15 percent are responding with a pledge. The new letter is more effective.


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Chapter EightOne-Sample Tests of One-Sample Tests of HypothesisHypothesis

Ka-fu Wong © 2003 Chap 8- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.

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