Aim: How do we use a t-test?

Exam FridayHW: complete worksheet

Recap: z-test

• Use a z-test when sample size is 30 or more

Example• A researcher claims that the average wind

speed in a certain city is 8 miles per hour. A sample of 32 days has an average wind speed of 8.2 miles per hour. The standard deviation of the sample is 0.6 mile per hour. At α=0.05, is there enough evidence to reject the claim? Use the P-value method.

Example Solution• Step 1: State the hypotheses and identify the claim

• Step 2: Compute the test value

• Step 3: Find the P-value (from z-table) – area under the curve

• Step 4: Make the decision– Since the P-value is greater than 0.05, the decision is not to rejected

the null hypothesis; therefore we ACCEPT the null hypothesis • Step 5: Summarize Results

– There is not enough evidence to reject the claim that the average wind speed is 8 miles per hour.

0

1

: 8

: 8

H

H

8.2 81.89

.6

32

Z

.5000 .4633 .0294

2(.0294) .0588P value

Drawing the Curve

Decision Rule when using P-values

• If P-value ≤ α, reject the null hypothesis.• If P-value > α do not reject the null hypothesis.

T-test• T-test: a statistical test for the mean of a

population and is used when the population is normally or approximately normally distributed and n < 30.

• The formula for the t test is

• The degree of freedom are d.f. = n - 1

Xt

s

n

Simple Example: Using Table F

• Find the critical t value of α = 0.05 with d.f. = 16 for a right tailed t test. – Solution:• Find the 0.05 column in the top row and 16 in the left-

hand column. • Where the row and column meet, the appropriate

critical value is found• It is +1.746

Hypothesis Testing With T-Test

• A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses’ salaries has a mean of $23,450 and a standard deviation of $400. Is there enough evidence to reject the director’s claim at α=0.05?

SolutionStep 1: State null and alternative hypothesis

Step 2: Find critical values- d.f. = 9, two tailed, α = 0.05, table F- critical values = ±2.262

Step 3: Find test value

Step 4: Make a decision- reject the null hypothesis since -4.35 < -2.262

Step 5: Summaries - There is not enough evidence to reject the claim that the starting salary of nurses is $24,000.

0

1

: $24,000

: $24,000

H

H

23450 240004.35

400

10

t

When to use what test?

• If sample standard deviation is known, use the z-test. The variable must be normally distributed if n < 30.

• If sample standard deviation is unknown but n ≥ 30, use the z test and use population standard deviation in formula.

• If sample standard deviation is unknown and n<30, use the t test. (The population must be approximately normally distributed.)

Class Work 8

• Worksheet• Must be handed in at the end of the period• NO LATE ASSIGNMENTS WILL BE ACCEPTED!