"Kind words can be short "Kind words can be short and easy to speak, and easy to speak,

but their echoes are truly but their echoes are truly endless“endless“

- Mother Teresa- Mother Teresa

Two Sample Means ProblemTwo Sample Means Problem

The board of directors at the Anchor Pointe Marina is The board of directors at the Anchor Pointe Marina is studying the usage of boats among its members. A studying the usage of boats among its members. A

sample of 30 members who have boats 10 to 20 sample of 30 members who have boats 10 to 20 feet in length showed that they used their boats an feet in length showed that they used their boats an

average of 11 days last July. The standard average of 11 days last July. The standard deviation of the sample was 3.88 days. For a deviation of the sample was 3.88 days. For a

sample of 40 member with boats 21 to 40 feet in sample of 40 member with boats 21 to 40 feet in length, the average number of days they used their length, the average number of days they used their boats in July was 7.67 with a standard deviation of boats in July was 7.67 with a standard deviation of

4.42 days. At the .02 significance level, can the 4.42 days. At the .02 significance level, can the board of directors conclude that those with the board of directors conclude that those with the smaller boats use their crafts more frequently?smaller boats use their crafts more frequently?

Step 1Step 1

State the null and alternative hypothesis.

H0: Large boat usage = small boat usage

H1: Smaller boat usage > large boat usage

Step 2Step 2

Select a level of significance.

This will be given to you. In this problem it is .02.

Step 3Step 3

Formulate a decision rule.

.5000 - .0200 = .4800 = 2.05z

Step 4Step 4Identify the test statistic.

11 – 7.67 = 3.35z

3.882 + 4.422

30 40

Step 5Step 5

Arrive at a decision.

The test statistic falls in the critical region, therefore we reject the null.

p-Value in Hypothesis Testingp-Value in Hypothesis Testing

• p-Value: The probability, assuming that the null hypothesis is true, of getting a value of the test statistic at least as extreme as the computed value for the test.

• If the p-value area is smaller than the significance level, H0 is rejected.

• If the p-value area is larger than the significance level, H0 is not rejected.

Statistical SignificanceStatistical Significance

p-Value: The probability of getting a sample outcome as far from what we would expect to get if the null hypothesis is true.

The stronger that p-value, the stronger the evidence that the null hypothesis is false.

Statistical SignificanceStatistical Significance

P-values can be determined by

- computing the z-score

- using the standard normal table

The null hypothesis can be rejected if the p-value is small enough.

P-ValueP-Value

1.64 Z 2.05Z

Tests Concerning Tests Concerning ProportionsProportions

Proportion:Proportion:

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

Tests Concerning Tests Concerning ProportionsProportions

• The sample proportion is denoted by p, where:

p = number of successes in the sample

number sampled

Test for One ProportionTest for One Proportion

π = population proportionp = sample proportion

Party, Party, Party!!!! Party, Party, Party!!!! Statistics is almost over. Statistics is almost over.

One Sample Proportion One Sample Proportion ProblemProblem

An urban planner claims that, nationally, An urban planner claims that, nationally, 20 percent of all families renting condos 20 percent of all families renting condos

move during a given year. A random move during a given year. A random sample of 200 families renting condos sample of 200 families renting condos in Dallas revealed that 56 had moved in Dallas revealed that 56 had moved

during the past year. At the .01 during the past year. At the .01 significance level, does this suggest significance level, does this suggest

that a larger proportion of condo that a larger proportion of condo owners moved in the Dallas area? owners moved in the Dallas area?

Determine the p-value.Determine the p-value.

Step 1Step 1

State the null and alternative hypothesis.

H0: Proportion = .20

H1: Proportion > .20

Step 2

Select a level of significance.

This will be given to you. In this problem, it is .01.

Step 3

Formulate a decision rule.

.5000 - .01 = .4900 = 2.32z

Step 4Identify the test statistic.

Z = .28 - .20 = 2.83z

.20(1-.20)

200

Step 5Arrive at a decision.

The test statistic falls in the critical region, therefore, we reject the null.

Test for Two ProportionsTest for Two Proportions

Two Proportion ProblemTwo Proportion ProblemSuppose that a random sample of 1,000 American-born citizens revealed that 198

favored resumption of full diplomatic relations with Cuba. Similarly, 117 of a sample of 500

foreign-born citizens favored it. At the .05 significance level, is there a difference in the proportion of American-born versus foreign-born citizens who favor restoring diplomatic

relations with Cuba?

Step 1Step 1

State the null and alternative hypothesis.

H0: Proportion of American-born = Foreign born

H1: Proportion of American-born ≠ Foreign-born

Step 2Step 2

Select a level of significance.

This will be given to you. In this problem it is .05.

Step 3Step 3Formulate a decision rule.

1.000 - .0500 = .9500

.9500/2 = .4750 = 1.96z

Step 4 – Part IStep 4 – Part IIdentify the test statistic.

PC = 198 + 117 = .21

1000 + 500

Step 4 – Part IIStep 4 – Part IIIdentify the test statistic.

Z = .198 - .234 = -1.61z

.21(1-.21) + .21(1-.21)1000 500

Step 5Step 5Arrive at a decision.

The test statistic falls in the null hypothesis region, therefore we fail to reject the null.

Type I and Type II ErrorsType I and Type II Errors

• Type I Error:Type I Error: • Rejecting the null hypothesis when H0 is actually true.

• Type II Error:Type II Error: • Accepting the null hypothesis when H0 is actually

false.

Type I ErrorType I Error

Rejecting the null hypothesis when H0 is actually true.

Type II ErrorType II Error

Accepting the null hypothesis when H0 is actually false.