Overview10.1 Inference for Mean Difference—

Dependent Samples

10.2 Inference for Two Independent Means

10.3 Inference for Two Independent Proportions

p-Value MethodMethod for carrying out hypothesis test

p-Value measures how well data fits null hypothesis.

p-ValueProbability of observing a sample statistic

(such as x or Zdata) at least as extreme as observed statistic assuming null hypothesis is true.

Roughly speaking, represents probability of observing sample statistic if the null hypothesis is true.

Since term “p-value” means “probability value,” always lies between 0 and 1.

Rejection Rule

When performing a hypothesis testusing the p-value method:

Reject H0 when the p-value is less than a.

Assessment of the Strength of Evidence

Table 9.5 Strength of evidence against the null hypothesis for various levels of p-value

10.1 Inference for Mean Difference—Dependent SamplesObjectives:By the end of this section, I will beable to…

1) Distinguish between independent samples and dependent samples.

2) Perform hypothesis tests for the population mean difference for dependent samples using the p-value method and the critical value method.

Independent Samples and Dependent SamplesTwo samples are independent when the

subjects selected for the first sample do not determine the subjects in the second sample.

Two samples are dependent when the subjects in the first sample determine the subjects in the second sample.

The data from dependent samples are called matched-pair or paired samples.

Example 10.1 - Dependent or independent sampling?Indicate whether each of the following experiments uses an independent or dependent sampling method.a. A study wished to compare the differences in price between name-brand merchandise and store-brand merchandise. Name-brand and store-brand items of the same size were purchased from each of the following six categories: paper towels, shampoo, cereal, ice cream, peanut butter, and milk.b. A study wished to compare traditional acupuncture with usual clinical care for a certain type of lower-back pain. The 241 subjects suffering from persistentnon-specific lower-back pain were randomly assigned to receive either traditional acupuncture or the usual clinical care. The results were measured at 12 and 24months.

Example 10.1 continuedSolution

a. For a given store, each name-brand item in the first sample is associated with exactly one store-brand item of that size in the second sample.

Items in the first sample determine the items in the second sample

Example of dependent sampling

b. Randomly assigned to receive either of the two treatments

Thus, the subjects that received acupuncture did not determine those who received clinical care, and vice versa.

Example of independent sampling.

Paired Sample t Test for the Population Mean: The Critical Value Method Step 1 State the hypotheses and the rejection rule.Use one of the hypothesis test forms from

Table 10.2 below. State clearly the meaning of μd.

Paired Sample t Test for the Population Mean: The p-value Method Step 2Enter the columns as lists in your calculator

and find the difference between the two columns.

Use the option STATS>TESTS>2:T-TEST.. Inpt: DATAChoose the list with the differences.Obtain the p-value.

Paired Sample t Test for the Population Mean: The Critical Value Method

Step 3 If p < α, reject the null hypothesis.

Step 4 State the conclusion and the interpretation.

10.2 Inference for Two Independent Means

Sampling Distribution of x1-x2

Random samples drawn independently from populations with population means μ1 and μ2 and either

Case 1: The two populations are normally

distributed, or

Case 2: The two sample sizes are large (at least 30),

then the quantity

Sampling Distribution of x1-x2 continued

Approximately a t distribution Degrees of freedom equal to the smaller of

n1 - 1 and n2 – 1x1 and s1 represent the mean and standard

deviation of the sample taken from population 1,

x2 and s2 represent the mean and standard deviation of the sample taken from population 2.

21

1 2 1 2 1 2 1 2

2 21 2

1 2

x x

x x x xt

s s sn n

Standard Error of x1-x2

Standard error of the statistic is

It measures the size of the typical error in using to measure μ1- μ2.

21

2 21 2

1 2x x

s ssn n

1 2x xs 1 2x x

1 2x x

Hypothesis Test for the Difference in Two Population Meansp-Value Method

Step 1 Select the option STATS>TESTS>4: 2-

SampTTEST.Enter the given information.Step 2 Find tdata.

Hypothesis Test for the Difference in Two Population Means continuedStep 3 Find the p-value using calculator.

Step 4 State the conclusion and interpretation. Compare the p-value with a.

10.3 Inference for Two Independent Proportions

Sampling Distribution of p1 - p2

Independent random samples from two populations

The quantity

1 2

1 2 1 2

ˆ ˆ

1 2 1 2

1 1 2 2

1 2

ˆ ˆ

ˆ ˆ

p p

p p p pZ

p p p pp q p qn n

Sampling Distribution of p1 - p2 continuedHas an approximately standard normal

distribution when the following conditions are satisfied:

x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5, (n2 - x2) ≥ 5

p1 and n1 represent the sample proportion and sample size of the sample taken from population 1 with population proportion p1;

Sampling Distribution of p1 - p2 continued

p2 and n2 represent the sample proportion and sample size of the sample taken from population 2 with population proportion p2;

q1 = 1 - p1 and q2 = 1 - p2.

Standard Error of p1 - p2

Standard error of the statistic p1 - p2

Where q1 = 1 - p1 and q2 = 1 - p2.

The standard error measures the size of the typical error in using p1 - p2 to estimate p1 - p2.

1 2ˆ ˆp ps

1 1 2 2

1 2ˆ ˆ

1 2

ˆ ˆ ˆ ˆp p

p q p qs

n n

1 2ˆ ˆp ps

Hypothesis Test for the Difference in Two Population Proportions: p-Value Method

Two independent random samplesTaken from two populationsPopulation proportions p1 and p2

Required conditions:

x1 ≥ 5, (n1 - x1) ≥ 5, x2 ≥ 5,

and (n2 - x2) ≥ 5.

Hypothesis Test for the Difference in Two Population Proportions: p-Value MethodStep 1State the hypotheses and the rejection rule

Use one of the forms from Table 10.19 page 576

State the meaning of p1 and p2

Reject H0 if the p-value is less than a.

Hypothesis Test for the Difference in Two Population Proportions: p-Value MethodStep 2 Find Zdata.

Zdata follows an approximately standard normal distribution if the required conditions are satisfied.

1 2

1 2

ˆ ˆ

1 1ˆ ˆ1data

pooled pooled

p pZ

p pn n

1 2

1 2

ˆ pooledx xpn n

Hypothesis Test for the Difference in Two Population Proportions: p-Value MethodStep 3 Find the p-value using calculator.

Step 4 State the conclusion and interpretation.