Lecture 5

A

Hypothesis

is a statement about the value of a

population parameter developed for the purpose

of testing. Examples of hypotheses made about a

population parameter are:

– The mean monthly income for systems analysts is $3,625.

• Hypotheses are tested in pairs.

• The alternative hypothesis is the claim that we

are trying to prove.

• The null hypothesis is the claim that we want

to disprove. The null hypothesis often

represents current belief about the value of

the parameter. We think that the current

Important Things to Remember about H

₀

and H

₁

• H

: null hypothesis and H

: alternate hypothesis

• H

and H

are mutually exclusive and collectively exhaustive

• H

is always presumed to be true

• H

has the burden of proof

• A random sample (n) is used to “reject H

”

• If we conclude 'do not reject H

', this does not necessarily mean

that the null hypothesis is true, it only suggests that there is not

sufficient evidence to reject H

; rejecting the null hypothesis then,

suggests that the alternative hypothesis may be true.

• Equality is always part of H

(e.g. “=” , “≥” , “≤”).

.

• Type I Error -

– Defined as the probability of rejecting the null

hypothesis when it is actually true.

– This is denoted by the Greek letter “



”

– Also known as the significance level of a test

• Type II Error:

– Defined as the probability of “accepting” the

null hypothesis when it is actually false.

.

• p-VALUE

is the probability of observing a sample value as

extreme as, or more extreme than, the value observed,

given that the null hypothesis is true.

• In testing a hypothesis, we can also compare the p-value

to with the significance level (



).

Steps in hypotheses testing

• Specify null and alternative hypotheses and select

• significance level

• Choose test statistic and formulate a decision rule

• Take a sample, make decision and interpret

Example:

A random sample of 17 police officers in Brownsville has a mean annual income of

$35,800 and a standard deviation of $7,800. In Greensville, a random sample of 18 police officers has a mean annual income of $35,100 and a standard deviation of $7,375. Test the claim at  = 0.01 that the mean annual incomes in the two cities are not the same. Assume the population variances are equal.

H_0 is not rejected

Two-Sample Tests of Hypothesis:

Dependent Samples

Dependent samples are samples that are paired or related in some fashion.

For example:

– If you wished to buy a car you would look at the same car at two (or more)

different dealerships and

compare the prices.

– If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program.

EXAMPLE

To perform a two-sample hypothesis test with dependent samples, the

difference between each data pair is first found:

d = x₁ – x₂

_{Difference between entries for a data pair.}

The test statistic is the mean of these differences.

d

t

d

s

n



/

Where

is the mean of the differences

sd is the standard deviation of the differences

n is the number of pairs (differences)

Step 1: State the null and alternate hypotheses.

H₀: _d = 0 H₁: _d ≠ 0

Step 2: State the level of significance.

The .05 significance level is stated in the problem.

Step 3: Find the appropriate test statistic.

We will use the t-test

Step 4: State the decision rule.

Reject H₀ if

t > t_{/2, n-1} or t < - t_/2,n-1

t > t_.025,9 or t < - t_{.025, 9}

t > 2.262 or t < -2.262

Step 5: Compute the value of t and make a decision

The computed value of t (3.305) is greater than the higher critical value (2.262), so our decision is to reject the null hypothesis.