a

Retain or Reject H0?

Outcome

a

Retain or Reject H0?

Outcome

a

Retain or Reject H0?

Outcome

a

Retain or Reject H0?

Outcome

a

Retain or Reject H0?

Outcome

a

Retain or Reject H0?

Outcome

a

a

Retain or Reject H0?

Outcome

a

The Binomial Distribution is a sampling distribution (for the sign test).

The sampling distribution of a statistic:(1) gives all the values that the statistic can take and (2) gives the probability of getting each value under the assumption that it

resulted from chance alone.

The sampling distribution is the chance distribution to which we compare our obtained results.

N = 10 Events

If chance alone is responsible, what is the probability of getting 5 tails? 6 tails? 9 tails? 7 or more tails?

.0001 .0001

N = 10 Events

H1= This coin is weighted for tails.H0= This coin is NOT weighted for tails.

Use an alpha of .05

.0001 .0001

N = 10 Events

H1= This coin is weighted.H0= This coin is NOT weighted.

Use an alpha of .05

Notice: The Binomial Test/Sign Test is ONLY appropriate when there are just two possibilities of outcome (for example heads/tails, correct/incorrect, etc….

Binomial Test Questions:1. Is a coin weighted if out of ten flips it ends up at heads 9 times?2. Does a fertilizer work if 9 times out of ten it produces greater growth than without it?3. Are college graduates smarter than non-college graduates if out of ten randomly sampled geniuses 9 of them happen to be college graduates?

THERE WILL BE A COMMON SET OF STEPS YOU WILL USE FOR ALL HYPOTHESIS TESTS:

Step 1: State your hypotheses.

Step 2: Calculate the critical value.

Step 3: Calculate the obtained statistic.

Step 4: Make a decision.

How about the following question:

It has been documented that in the general population, the average number of hours slept a night by Americans is 8, with a standard deviation of 4 hours. I collect sample data on a set of 16 college students, and determine that their average number of hours sleeping per night is 6 hours. Do college students on average sleep less than the general population?

In other words, I am comparing a sample mean (my 16 students) with my KNOWN population mean and asking the question, are they different?

It has been documented that in the general population, the average number of hours slept a night by Americans is 8, with a standard deviation of 4 hours. I collect sample data on a set of 16 college students, and determine that their average number of hours sleeping per night is 6 hours (s=1). Do college students on average sleep less than the general population?

In other words, I am comparing a sample mean (my 16 students) with a population mean and asking the question, are they different?

8

m = 8s = 4

6

population

samplex = 6s = 1

78892

66666

667711

858104

704610

53596

24063

383148

102195

175711

53598

45943

592412

1214107

711766

What is a sampling distribution of the mean?

8

m = 8s = 4

78892

66666

667711

858104

704610

53596

24063

383148

102195

175711

53598

45943

592412

1214107

711766 …

What is a sampling distribution of the mean?

8

m = 8s = 4

It has been documented that in the general population, the average number of hours slept a night by Americans is 8, with a standard deviation of 4 hours. I collect sample data on a set of 16 college students, and determine that their average number of hours sleeping per night is 6 hours (s=1). Do college students on average sleep less than the general population?

In other words, I am comparing a sample mean (my 16 students) with a population mean and asking the question, are they different?

8

m = 8s = 4

6

population

samplex = 6s = 1

What is a sampling distribution of the mean?

78892

66666

667711

858104

704610

53596

24063

383148

102195

175711

53598

45943

592412

1214107

711766

6.8 6 7.4 7 5.4 5.6 3 7.2 5.4 6.2 6 5 6.4 6.8 7.4

8

mx = 8sx = 4/sqrt(16)

_

_

412352

5.2

x_

= 5.75

What is a sampling distribution of the mean?

78892

66666

667711

858104

704610

53596

24063

383148

102195

175711

53598

45943

592412

1214107

711766

8

mx = 8sx = 4/sqrt(16)

_

_

412352…

x_

= 5.75= 5.63= …= …= …

What is a sampling distribution of the mean?

78892

66666

667711

858104

704610

53596

24063

383148

102195

175711

53598

45943

592412

1214107

711766

8

mx = 8sx = 4/sqrt(16)

_

_

412352…

x_

= 5.75= 5.63= …= …= …

Sampling and Variation

Sample Size & Estimated Nail Biting at closing time by Stock Brokers

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Nail Biting

Freq

uenc

y n=1000n=100n=10

It has been documented that in the general population, the average number of hours slept a night by Americans is 8, with a standard deviation of 4 hours. I collect sample data on a set of 16 college students, and determine that their average number of hours sleeping per night is 6 hours (s=1). Do college students on average sleep less than the general population?

In other words, I am comparing a sample mean (my 16 students) with a population mean and asking the question, are they different?

8 6

Sampling distribution (the distribution of means)

samplex = 6s = 1

mx = 8sx = 4/√16

_

_

The z-test• When comparing a sample mean to a population mean

AND THE POPULATION STANDARD DEVIATION IS KNOWN, use a z-test.

x

xobt

xz

Standard error of the mean and z-scores

sx = n

z-score for a sample meanx -mx

Zobt = sx

Introducing the Z-test

Calculate the obtained statistic.

Make a decision.

H1: College students sleep less than people in the general population.

H0: College students DO NOT sleep less than people in the general population.

We need to find the zcrit for a one-tailed test (alpha = .05).

Using the z-table, we can find: zcrit = -1.65

zobt = (6-8)/1 = -2

Reject the null. College students do indeed sleep less than people in the general population.

State your hypotheses:

Calculate the critical value:

zobt = (x-mx) /sx

_

Another z-test example:

In the population, individuals spend on average $10/day on alcohol, with a standard deviation of $2. I suspect that among graduate students, the average amount spent is greater than the population average. I gather together 25 graduate students from this program and calculate the average amount spent on alcohol/day: $11.00 (s= $1). Use an alpha level of .05.

How can I test whether or not graduate students differ significantly from the general population in this regard?

How do I know I need to be using a z-test?

1. Identify your null and alternative hypotheses:

2. Determine the critical region of the sampling distribution in which you will reject the null hypothesis:

3. Calculate the test statistic:

4. Make a decision about your null hypothesis:

H1: College students spend more money on alcohol per day than a person from the general population.

H0: College students DO NOT spend more money on alcohol per day than a person from the general population.

Since we are using a z-test, the z-distribution is our sampling distribution. In order to find Zcrit, we must know about the number of tails of our hypothesis, as well as the alpha level. Alpha = .05, Tails = 1.

Zcrit = 1.65 (This means that if our Zobt value exceeds 1.65, only then will we reject the null).

Zobt = 11 - 10

.40

Reject the null.

= 2.5We know that: Zobt = x – mx

sx

_

Yet another z-test example:

In a population of American graduate students, individuals earn on average $7.25/hour, with a standard deviation of $5. I want to know whether or not graduate students from Brooklyn earn a different wage than grad students in general. I gather together 9 graduate students from this program and calculate the average amount they earn an hour: $11.00 (s= $.50). Use an alpha level of .01.

How can I test whether or not graduate students differ significantly from the general population in this regard?

How do I know I need to be using a z-test?

1. Identify your null and alternative hypotheses:

2. Determine the critical region of the sampling distribution in which you will reject the null hypothesis:

3. Calculate the test statistic:

4. Make a decision about your null hypothesis:

H1: Graduate students in Brooklyn earn a different amount than graduate students in general.

H0: Graduate students in Brooklyn DO NOT earn a different amount than graduate students in general.

Since we are using a z-test, the z-distribution is our sampling distribution. In order to find Zcrit, we must know about the number of tails of our hypothesis, as well as the alpha level. Alpha = .01, Tails = 2.

Zcrit = -2.58 and 2.58 (This means that if our Zobt value exceeds either -2.58 or 2.58, only then will we reject the null).

Zobt = 11 - 7.25

1.67

Retain the null.

= 2.25 We know that: Zobt = x – mx

sx

_

What do these problems we have been working on have in common?

In a population of American graduate students, individuals earn on average $7.25/hour on alcohol, with a standard deviation of $5. I jokingly ask whether or not graduate students from Brooklyn earn a different wage than grad students in general. I gather together 9 graduate students from this program and calculate the average amount they earn an hour: $11.00 (s= $.50). Use an alpha level of .01.

How do I know I need to be using a z-test?

1. We are comparing a sample mean to a KNOWN population mean.

2. We KNOW the population s.

1. We are comparing a sample mean to a KNOWN population mean.1. We are comparing a sample mean to a KNOWN population mean.

2. We KNOW the population s.