Introduction to Hypothesis Testing

The purpose of hypothesis testing is to determine whether there is enough statistical evidence supporting a certain belief or claim about a parameter.. Examples • Is there statistical

The purpose of hypothesis testing is to determine

whether there is enough statistical evidence supporting

a certain belief (or claim) about a parameter.

Examples

• Is there statistical evidence that support the

hypothesis that more than p% of all potential customers will purchase a new products?

• Is the hypothesis that a certain drug is effective?

Step 1: Two hypotheses are defined.

H 0 : The null hypothesis specifies our current belief

about the parameter we test (m = 170, p = 4, etc.) Must be a specific value.

H 1 : The alternative hypothesis specifies a range of

values for the parameter tested (m > 170; p ¹ 4; etc.) effected by the belief.

11.1 Concepts of Hypothesis Testing

The Reject-Region method

H 0 Needed in the test Do not reject H 0

Claim H 0 H A

m=A m=A m≠A 2 –tail

or 2-side test m>A m=A m>A 1 –tail

or 1-side test m≥A m=A m<A 1 –tail

or 1-side test m<A m=A m<A 1 –tail

or 1-side test m≤A m=A m>A 1 –tail

or 1-side test

m=A m=A m≠A 2 -tail

or 2-side test m>A m=A m>A 1 –tail

or 1-side test m≥A m=A m<A 1 –tail

or 1-side test m<A m=A

m≤A m=A

m=A m=A m≠A 2 -tail

or 2-side test m>A m=A m>A 1 –tail

or 1-side test m≥A m=A

Example 1: The manager of a department determines

that new billing system will be cost-effective only if the mean monthly account is more than $170 A random

sample of 400 monthly accounts is drawn, for which the sample mean is $178 Assume a standard deviation of

$65 Can the manager conclude from this that the new system will be cost-effective?

H 1 : µ > 170

10.2 Testing the Population Mean when the Population Standard Deviation is Known

accept H 0 OK Type II error

b= = Probability of committing the type II error

The more severe the consequence of committing the

Send an innocent person

Send an innocent person

Step 3: Determine the sample size n and hence the

sampling distribution.

Example 1:n=400; N(0,1)

Step 4: Depending on the sampling distribution , the H A ,

the reject region

H A : μ > 170 (1-sided test)

1.645 Critical value

N(0,1)

a =.05

Reject region: Z > 1.645

Right-Tail Testing

Left-Tail Testing

Two–Tail Testing

Step 5: Collect data x = 178

Calculate the standard test statistic

46

2 400

/ 65

Reject H 0 : There is enough statistical evidence to

conclude that the alternative hypothesis is true.

evidence to conclude that the alternative hypothesis is true.

Step 6: If the test statistic is in the reject region, then

A Left Hand Tail Test

Reject H0 if falls here x Critical

value

The SSA envelop plan example: The chief financial

officer in FedEx believes that including a stamped addressed (SSA) envelop in the monthly invoice sent

self-to cusself-tomers will decrease the amount of time it take for customers to pay their monthly bills Currently,

customers return their payments in 22 days on the

average, with a standard deviation of 6 days.

A random sample of 220 customers was selected and SSA envelops were included with their invoice packs The mean time it took customers to pay their bill was 21.63

Can the CFO conclude that the plan will be successful

at 10% significance level?

/ 6

22 63

Critical value Reject H0 if falls here x

Assuming the standard deviation of the bills of the

competitors is 3.87, can we infer that there is a

difference between AT&T’s bills and the competitor’s

bills (on the average)?

55

17

=

x

1 100

/ 87

3

09

17 55

1.96 Critical value