bcor 1020 business statistics lecture 23 – april 15, 2008

BCOR 1020Business Statistics

Lecture 23 – April 15, 2008

Overview

• Chapter 10 – Two Sample Tests– Comparing Two Means (s unknown)– Paired Comparisons

Chapter 10 – Comparing Two Independent Means (s unknown)

• Just as when the standard deviations are known, the hypotheses for comparing two independent population means 1 and 2 are:

Format of Hypotheses:


• If the population variances 12 and 2

2 are known, then we use a standard normal distribution (Tuesday’s notes).

• If the population variances 12 and 2

2 are unknown, then we use a student’s t distribution.

• There are three possible cases…

Test Statistic:

Case 1: Known Variances (Last Lecture)

2

22

1

21

21

nn

xxZ

Use the standard normal table to

define the rejection region or calculate the p-value.


• Since the variances are unknown, they must be estimated and the Student’s t distribution used to test the means.

• Assuming the population variances are equal, s1

2 and s22 can be used to estimate a common

pooled variance sp2.

Case 2: Unknown Variances, Assumed Equal

• The test statistic is With degrees of

freedom

= n1 + n2 – 2.


Case 3: Unknown Variances, Assumed Unequal

• If the unknown variances are assumed to be unequal, they are not pooled together.

• In this case, the distribution of the random variable x1 – x2 is not certain.

• Use the Welch-Satterthwaite test which replaces 12

and 22 with s1

2 and s22 in the known variance Z*

formula, then uses a Student’s t test with adjusted degrees of freedom.


Case 3: Unknown Variances, Assumed Unequal

• Welch-Satterthwaite test

• with degrees of freedom

• A Quick Rule for degrees of freedom is to use ~ min(n1 – 1, n2 – 1).


• Choose the level of significance, .• Choose the appropriate hypotheses• Calculate the Test Statistic• State the decision rule – Based on , determine the

critical value(s).

Steps in Testing Two Means:

• Make the decision – Reject H0 if the test statistic falls in the rejection region(s) as defined by the critical value(s).

For example, for a two-tailed test for Student’s t and = .05


• If the sample sizes are equal, the Case 2 and Case 3 test statistics will be identical, although the degrees of freedom may differ.

• If the variances are similar, the two tests will usually agree.

• If no information about the population variances is available, then the best choice is Case 3.

• The fewer assumptions, the better – When in doubt, use Case 3!

When to Use Each Case Statistic:

Must Sample Sizes Be Equal?• Unequal sample sizes are common and the formulas

still apply.


Example:• A restaurant chain is considering closing one of two stores.

– In a sample of 16 randomly selected days, restaurant A has average daily sales of $3000 with a standard deviation of SA = $450.

– In a sample of 12 randomly selected days, restaurant B has average daily sales of $2700 with a standard deviation of SB = $400.

• All other things being equal, the restaurant with lower sales will be closed.

• Assuming that sales for restaurants within the same chain will have equivalent standard deviations, test the appropriate hypothesis to determine whether sales at restaurant B are significantly lower than sales at restaurant A at the 5% level of significance.

(Overhead)


Example (continued):• Assumptions:

– Sales data are independent– The standard deviations are unknown, but assumed equal.

• We will treat this as Case 2:

• Hypotheses: we want to determine if A > B…

H0: A < B vs. H1: A > B (Right-tail test)

• Test Statistic: (using pooled variance sp2)…


Example (continued):• Test Statistic: (using pooled variance sp

2)…

2308.18451926

4797500

21216

400)112(450)116(

2

)1()1( 22222

BA

BBAAp nn

snsns

121

16122

*

2308.184519

27003000

B

p

A

p

BA

n

S

n

S

xxT

829.1* T t distribution with = 16 + 12 – 2 = 26 d.f. under H0.

Clickers

What is the rejection criteria for this problem?

(A) Reject H0 in favor of H1 if T* > 1.645.

(B) Reject H0 in favor of H1 if T* > 1.706.

(C) Reject H0 in favor of H1 if T* > 1.701.

(D) Reject H0 in favor of H1 if T* > 2.056.

(E) Reject H0 in favor of H1 if T* > 1.315.

Clickers

What is your decision?

(A) Reject H0 in favor of H1.

(B) Fail to Reject H0 in favor of H1.

(C) Not enough information.


Example (conclusion):• Since our test statistic, T*, falls in the rejection region,

we will reject H0 in favor of H1.

• Based on the data collected, there is statistically significant evidence that A > B.

• So, all other things being equal, we will close restaurant B.


Example (revisited):• How does our test change if we don’t assume equal

variances (Case 3)?• We will use the same hypothesis test:

H0: A < B vs. H1: A > B (Right-tail test)

• With a different test statistic…

12400

1645022

*

22

27003000

B

B

A

A

BA

nS

nS

xxT

861.1* T T distribution with ~ min(nA – 1,nB – 1) = min(16 – 1, 12 – 1) = 11 d.f. under H0.


Example (revisited):• Rejection Criteria:

For the right-tail test, we will reject H0 in favor of H1 if T* > t,.

• Decision: Since T* = 1.861 > t, = t.05,11 = 1.796, we will reject H0 in favor of H1.

• Just as before, based on the data collected, there is statistically significant evidence that A > B.

• Our exact p-value will be a little larger in this case since this test makes fewer assumption – and is therefore more conservative.

Chapter 10 – Paired Comparisons (Dependent Samples)

Paired Data:• Data occurs in matched pairs when the same item is

observed twice but under different circumstances.• For example, blood pressure is taken before and

after a treatment is given.• Paired data are typically displayed in columns:


Paired t Test:• Paired data typically come from a before/after

experiment on n subjects – so the n observations of the two variable are dependent.

• In the paired t test, the difference between x1 and x2 is measured as d = x1 – x2.

• The mean d and standard deviation sd of the sample of n differences are calculated with the usual formulas for a mean and standard deviation.


Paired t Test:• The calculations for the mean and standard

deviation are:

• Since the population variance of d is unknown, use the Student’s t with n – 1 degrees of freedom. n

Sd

d

dT

*


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: d >

H1: d <

(ii) H0: d <

H1: d >

(iii) H0: d =

H1: d

• If the null hypothesis is true and d = 0, then T* has the student’s t distribution with n – 1 d.f.

Decision Criteria: (the same as any other t-test)• We can either compare T* to a critical value of the

appropriate t distribution or calculate (bound) the p-value for the test.


Example:• A new cell phone battery is being considered as a

replacement for the current one. Six college students were selected to try each battery in their usual mix of “talk” and “standby” and to record the number of hours until recharge was needed. The data is below. Using a level of significance of a = 5%, do these results show that the newer battery has significantly longer life?

Student 1 2 3 4 5 6

New Battery 41 53 40 43 49 43

Old Battery 34 40 38 44 44 33

(Overhead)di = New - Old 7 13 2 -1 5 10


Example (continued):• We can calculated the sample mean and standard

deviation for the dis…

14.5ds

614.5

* 000.6

n

Sdd

d

xT

86.2* T

00.6dx and

• Based on the problem statement, we will test the hypothesis H0: d < 0 vs. H1: d > 0 (which corresponds to the battery life for the new battery being greater).

• The test statistic is…

ort distribution with = n – 1 = 5 d.f. under H0.


Example (continued):• Rejection Criteria:

For the right-tail test, we will reject H0 in favor of H1 if T* > t,.

• Decision: Since T* = 2.86 > t, = t.05,5 = 2.015, we will reject H0 in favor of H1.

• Based on the data collected, there is statistically significant evidence that the new battery lasts longer.

Clickers

For this paired t test, our test statistic T* = 2.86 has

a student’s t distribution with = 5 degrees of freedom. Use the t-table to find appropriate bounds on the p-value of this test.

(A) 0.01 < p-value < 0.02

(B) 0.02 < p-value < 0.025

(C) 0.025 < p-value < 0.05

(D) 0.05 < p-value < 0.10

bcor 1020 business statistics lecture 23 – april 15, 2008

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