BCOR 1020Business Statistics

Lecture 20 – April 3, 2008

Overview

• Chapter 9 – Hypothesis Testing– Problem: Testing A Hypothesis on a

Proportion– Testing a Mean (): Population Variance ()

Known– Problem: Testing a Mean () when Population

Variance () is Known

Problem: Testing A Hypothesis on a Proportion

Suppose we revisited our earlier example and conducted an additional survey to determine whether the proportion of our target market that is willing to pay $25 per unit exceeds 20% (as required by the business case).• In our new survey, 72 out of 300 respondents said they

would be willing to pay $25 per unit.• At the 5% level of significance, conduct the appropriate

hypothesis test to determine whether the population proportion exceeds 20%.

• Include the following:– State the level of significance, .– State the null and alternative hypotheses, H0 and H1.– Compute the test statistic– State the decision criteria– State your decision

(Overhead)

Problem: Testing A Hypothesis on a Proportion

Work: =

H0:

H1:

Test Statistic (and Distribution under H0)…

Decision Criteria…

Decision…

Clickers

What are the appropriate null and alternative hypotheses?

(B) H0: >

H1: <

(A) H0: <

H1: >

(C) H0: =

H1:

Clickers

What is the point estimate of the population proportion ?

(A) p = 0.15

(B) p = 0.20

(C) p = 0.22

(D) p = 0.24

(E) p = 0.36

Clickers

What is the calculated value of your test statistic?

(A) Z* = 2.00

(B) Z* = 1.73

(C) Z* = 0.87

(D) Z* = 0.71

Clickers

What is your decision criteria?

(A) Reject H0 if Z* < -1.645

(B) Reject H0 if Z* < -1.960

(C) Reject H0 if Z* > 1.960

(D) Reject H0 if Z* > 1.645

(E) Reject H0 if |Z*| > 1.960

Problem: Testing A Hypothesis on a Proportion

Conclusion:• Since our test statistic Z* = 1.73 > Z = 1.645,

we will reject H0 in favor of H1: > 0.20.

• Based on the data in this sample, there is statistically significant evidence that the population proportion exceeds 20%.

Chapter 9 – Testing a Mean ( known)

Hypothesis Tests on ( known):• If we wish to test a hypothesis about the mean of a

population when is assumed to be known, we will follow the same logic…

– Specify the level of significance, (given in problem or assume 10%).

– State the null and alternative hypotheses, H0 and H1 (based on the problem statement).

– Compute the test statistic and determine its distribution under H0.

– State the decision criteria (based on the hypotheses and distribution of the test statistic under H0).

– State your decision.

Chapter 9 – Testing a Mean ( known)

Selection of H0 and H1:• Remember, the conclusion we wish to test should be

stated in the alternative hypothesis.• Based on the problem statement, we choose from…

(i) H0: > 0

H1: < 0

(ii) H0: < 0

H1: > 0

(iii) H0: = 0

H1: 0

where 0 is the null hypothesized value of (based on the problem statement).

Chapter 9 – Testing a Mean ( known)

Test Statistic:• Start with the point estimate of , in xx 1

• Recall that for a large enough n {n > 30}, is approximately normal with x n

x

xand

• If H0 is true, then is approximately normal with

0 x nx

and

x

• So, the following statistic will have approximately a standard normal distribution:

n

xZ

0*

Chapter 9 – Testing a Mean ( known)

Decision Criteria:• Just as with tests on proportions, our decision criteria will

consist of comparing our test statistic to an appropriate critical point in the standard normal distribution (the distribution of Z* under H0).

(i) For the hypothesis test H0: > 0 vs. H1: < 0, we will reject H0 in favor of H1 if Z* < – Z.

(ii) For the hypothesis test H0: < 0 vs. H1: > 0, we will reject H0 in favor of H1 if Z* > Z.

(iii) For the hypothesis test H0: = 0 vs. H1: 0, we will reject H0 in favor of H1 if |Z*| > Z/2.

Chapter 9 – Testing a Mean ( known)

Example (original motivating example):• Suppose your business is planning on bringing a

new product to market.• There is a business case to proceed only if

– the cost of production is less than $10 per unit

and– At least 20% of your target market is willing to pay $25

per unit to purchase this product.

• How do you determine whether or not to proceed?

Chapter 9 – Testing a Mean ( known)

Motivating Example (continued) :• Assume the cost of production can be modeled

as a continuous variable.– You can conduct a random sample of the

manufacturing process and collect cost data.– If 40 randomly selected production runs yield and

average cost of $9.00 with a standard deviation of $1.00, what can you conclude?

– We will test an appropriate hypothesis to determine whether the average cost of production is less than $10.00 per unit (as required by the business case).

(Overhead)

Chapter 9 – Testing a Mean ( known)

Motivating Example (continued) :• Conduct the Hypothesis Test:

– Specify , say = 0.05 for example.– Select appropriate hypotheses.

• Since we want to test that the mean is less than $10, we know that 0 = 10 and H1 should be the “<“ inequality.

• So we will test (i) H0: > 0 vs. H1: < 0.

– Calculate the test statistic…

401

0* 109

n

xZ

32.6* Z

Chapter 9 – Testing a Mean ( known)

Motivating Example (continued) :• Conduct the Hypothesis Test:

– Use the Decision criteria:

• (i) we will reject H0 in favor of H1 if Z* < – Z, where Z.05 = 1.645.

– State our Decision…

• Since Z* = -6.32 < -Z.05 = -1.645, we reject H0 in favor of H1.

– In “plain” language…• There is statistically significant evidence that the

average cost of production is less than $10 per unit.

Chapter 9 – Testing a Mean ( known)Calculating the p-value of the test:• The p-value of the test is the exact probability of

a type I error based on the data collected for the test. It is a measure of the plausibility of H0.– P-value = P(Reject H0 | H0 is True) based on our data.– Formula depends on which pair of hypotheses we are testing…

*2 ZZPvaluep

(i) For the hypothesis test H0: > 0 vs. H1: < 0,

(ii) For the hypothesis test H0: < 0 vs. H1: > 0,

(iii) For the hypothesis test H0: = 0 vs. H1: 0,

)( *ZZPvaluep

)( *ZZPvaluep

Chapter 9 – Testing a Mean ( known)

• Example: Let’s calculate the p-value of the • test in our example…• We found Z* = –6.32

• Since we were testing H0: > 0 vs. H1: < 0,

0)32.6()( * ZPZZPvaluep

Interpretation:If we were to reject H0 based on the observed data, there is approximately zero probability that we would be making a type I error. Since this is smaller than = 5%, we will reject H0.

Problem: Testing A Hypothesis on a Mean ( Known)

In an effort to get a loan, a clothing retailer has made the claim that the average daily sales at her store exceeds $7500. Historical data suggests that the purchase amount is normally distributed with a standard deviation of = $1500.• In a randomly selected sample of 24 days sales data, the

average daily sales were found to be = $7900.• At the 10% level of significance, conduct the appropriate

hypothesis test to determine whether the data supports the retailer’s claim.

• Include the following:– State the level of significance, .– State the null and alternative hypotheses, H0 and H1.– Compute the test statistic– State the decision criteria– State your decision

x

(Overhead)

Problem: Testing A Hypothesis on a Mean ( Known)

Work: =

H0:H1:

Test Statistic (and Distribution under H0)…

Decision Criteria…

Decision…

Clickers

What are the appropriate null and alternative hypotheses?

(A) H0: >

H1: <

(B) H0: <

H1: >

(C) H0: =

H1:

Clickers

What is the calculated value of your test statistic?

(A) Z* = 0.27

(B) Z* = 1.28

(C) Z* = 1.31

(D) Z* = 1.96

(E) Z* = 5.33

Clickers

What is your decision criteria?

(A) Reject H0 if Z* < -1.645

(B) Reject H0 if Z* < -1.282

(C) Reject H0 if Z* > 1.282

(D) Reject H0 if Z* > 1.645

(E) Reject H0 if |Z*| > 1.960

Clickers

What is your decision?

(A) Reject H0 in favor of H1

(B) Fail to reject (Accept) H0 in favor of H1

(C) There is not enough information

Problem: Testing A Hypothesis on a Mean ( Known)

Conclusion:• Since our test statistic Z* = 1.31 is greater than

Z = 1.282, we will reject H0 in favor of H1: > $7500.

• Based on the data in this sample, there is statistically significant evidence that the average daily sales at her store exceeds $7500.

Clickers

Given our test statistics Z* = 1.31, calculate the p-value for the hypothesis test H0: < 7500 vs. H1: > 7500.

(A) p-value = 0.0060

(B) p-value = 0.0951

(C) p-value = 0.9940

(D) p-value = 0.9049