Inferences About Means of Single Samples

Chapter 10

Homework: 1-6

Evaluating Hypotheses About Means

Evaluating hypothesis about population simplest research situation taking a single sample

Test statistics z test if known t test if unknown

If reject H0 evaluate practical significance ~

Steps in Hypothesis Evaluation

1. State null & alternative hypotheses

2. Set criterion for rejecting H0

3. collect sample; compute sample statistic & test statistic

4. Interpret results Steps 1 & 2 before collecting data ~

1. State null & alternative hypotheses

unknown calculate X, s, & s X from sample use t test

Average college students study 21 hours per week? Do Coe students study 21 hrs/week?

Nondirectional hypothesis: n = 16 H0 : = 21; H1 : 21

reject H0 if increase or decrease ~

1. State null & alternative hypotheses

3 important distributions variable: X sample statistic: X

central limit theorem test statistic: z or t

known probabilities Distributions show all possible values of

variable assuming H0 is true

0 +1 +2-2 -1

What does distribution of sample statistic look like if Ho true?

If Ho is false?

1. State null & alternative hypotheses

Test statistic General form

test statistic = sample statistic - population parameter

standard error of sample statistic

2. Set Criterion for Rejecting H0

Directionality & level of significance Xobs = computed sample statistic

same as randomly drawing a single X from sampling distribution of means

X CV = critical value of the statistic set in advance beginning of the rejection region

area in tails of distribution

if Xobs lies beyond, reject H0 ~

2. Set Criterion for Rejecting H0

Computing critical values of statistic X CV = + tCV (sX) same as confidence intervals X CV = upper & lower limits

reject H0 if beyond *Critical value of test statistic

df = 15 t.05 = 2.131 (nondirectional) ~

2. Set Criterion for Rejecting H0

Rejection region portion of distribution beyond critical

valuearea in tails

for sample statistic or test statistic Level of significance

if = .05 nondirectional: .025 in each tail

.025 + .025 = .05 ~

Rejection regions

f

1 20-1-2

+2.131-2.131

2. Set Criterion for Rejecting H0

Test statistic observed value

computed from sample critical value

• criterion set in advance• depends on (level of significance)• & directionality

• nondirectional: t.05 = 1.96

• if directional: t.05 = 1.645 ~

X

obs s

Xt

2. Set Criterion for Rejecting H0

Decision if tobs is beyond tCV,then reject H0

if not, “accept” H0

3. Collect sample & compute statistics

Collect data & compute test statistic X = 24.63; s = 7.78, s X = 1.94

Test statistic

X

obs s

Xt

94.1

2163.24 87.1

94.1

63.3

4. Interpret Results Is tobs is beyond tCV?

NO. 1.87 < 2.131 then “accept” H0

Students study about 21 hrs per week. No significant difference

does not mean they are equal not sufficient data to reject

Practical significance not an issue ~

A Directional Hypothesis

unknown: same question evidence from prior surveys that

Coe students study more than 21 hrs per week

experimental hypothesis = H1 can use directional hypotheses

A Directional Hypothesis

1. State H0 & H1

H0: < 21

Coe students study less than or equal to 21 hrs per week

H1: > 21

Coe students study more than 21 hrs per week ~

A Directional Hypothesis

2. Set criterion for rejecting H0

= .05, level of significance directional (one-tailed) test df = 15 tCV = 1.753

critical value for area = .05 (one-tailed) ~

A Directional Hypothesis

3. Collect sample & compute statistics X = 24.63; s = 7.78, s X = 1.94

test statistic = tobs

X

obs s

Xt

94.1

2163.24 87.1

94.1

63.3

A Directional Hypothesis

3. Interpret results tobs > tCV

1.87 > 1.753 Reject H0

accept H1

Coe students study more than 21 hours per week ~

Practical Significance

Statistical significance? YES

Practical significance? MAYBE

Determining practical significance effect size ~

Practical Significance: Effect size

Magnitude of the result (difference) Raw effect size

measured on scale of original data Xobs - = 24.63 - 21 = 3.63 Coe students study 3.63 hours per

week longer than the national average ~

Practical Significance: Effect size Effect size index

compare effect size for variables using different scales (e.g. GRE, ACT)

divide difference by s nondirectional directional

s

Xd

obs

s

Xd

obs

47.78.7

2163.24

d

0 +1 +2-2 -1

d = .47 standard deviations above the mean

Practical Significance: Effect size

Is effect magnitude practically significant? .5 considered moderate effect size

e.g., Is it worth using a new statistics textbook that test scores d = .5? Ultimately we must make decision using our expertise considering many factors ~

When Is Known

Usually not the situation calculate X from sample use z test degrees of freedom not relevant find zCV in z table

use X

X

obs

Xz

Practical Significance: Effect size Effect size index: is known

nondirectional directional

obsXd

obsX

d

Relationship to Confidence Intervals

Nondirectional tests equivalent to CI Level of significance: = .05 Level of confidence: 1- = .95

95% confident that true value of falls within interval

if it does: H0 is true

If falls outside CI: reject H 0 ~