Chapter 9Chapter 9Tests of SignificanceTests of Significance

Target Goal: Target Goal: I can perform a significance test to support the I can perform a significance test to support the alternative hypothesis.alternative hypothesis.I can interpret P values in context.I can interpret P values in context.

9.1a Testing a Claimh.w: pg. 546: 1-7 odd

Tests of Significance: Tests of Significance: to test a claim about an to test a claim about an unknown population parameter.unknown population parameter.Example :Example : Sweetening Colas Sweetening Colas

• Diet colas use artificial sweeteners to avoid sugar. Colas with artificial sweeteners gradually lose sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then stored for a period of time, then each taster scores the stored cola.

• This is a matched pairs experiment.

• The reported data is the difference in tasters’ scores. The bigger the difference, the bigger the loss in sweetness.

2.0 0.4 0.7 2.0 -0.42.2 -1.3 1.2 1.1 2.3

Positive score: lost sweetnessNegative score: gained sweetness• The sample mean indicates a small loss of

sweetness. 1.02x

• Consider that a different sample of tasters would have resulted in different scores, and that some variation in scores is expected due to chance.

Does the data provide good evidence that the cola lost sweetness in storage?

Perform a Perform a Significance TestSignificance Test..

Step 1: State - Identify the population parameter. State the null and alternative hypothesis.

• The parameter of interest is μ, the mean loss in sweetness of all colas produced by the manufacturer.

.

There is no effect or change in the population.

This is the statement we are trying to find evidence against.

• The cola does not lose sweetness.

“ H- nought or null hypothesis”

There is an effect or change in the population.• This is the statement we are trying to find

evidence for.• The cola does lose sweetness.

alternative hypothesis : 0aH

0 : 0H

Step 2: Step 2: PlanPlan

Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

• We will be more specific about this later.

Step 3: Step 3: Do - Do - Calculate Calculate a statistica statistic to estimate the parameter.to estimate the parameter.

Is the value of the statistic far from the value of the parameter?

• If so, reject the null hypothesis.

• If not, accept the null hypothesis.

Calculate the test statistic.Calculate the test statistic.• Suppose the individual tasters’ scores vary

according to a normal distribution with mean μ and σ = 1.

• We want to test the null hypothesis so we assume μ = 0.

• The sampling model for is approximately normal with mean and standard deviation 1

0.31610n

x0

N(0, 0.316)N(0, 0.316) Compare one cola with = 0.3 and our cola with = 1.02 to show what it means in terms of .

-0.9 -0.6 -0.3 0 0.3 0.6 0.9

The cola with 0.3 could happen just by chance.Our cola is so far out on the normal curve that it is good evidence that this cola did lose sweetness.Reject the null hypothesis.

xx

0 : 0H

Our sample mean, , was 1.02. • Assuming that the null hypothesis is true, what

is the probability of getting a result at least that large?

Calculate the P-value. (the probability of the observed )• If P-value is small, your result is statistically

significant.

x

x

Determine Determine P-valueP-value

• Normalcdf ( 1.02, 1E99, 0, 0.316 ) =• The probability to the right of is called the

P-value. • The P-value is small, so our result is statistically

significant. Reject the null hypothesis.

1.02

0.0006

0.0006

x

Alternate way to calculate one sided P-ValueAlternate way to calculate one sided P-ValueFind Z-value for N(0, 0.316 ), Find Z-value for N(0, 0.316 ), for sample mean 0.3.for sample mean 0.3.

• = P(Z ≥ 0.95)• = 1 - P(Z ≤ 0.95)• = 1 – 0.8289 = 0.1711

Is this good evidence against the null hypothesis?

( 0.3)P x 0 0.3 0

0.316 0.316

xP

• There is about a 17% chance that we will obtain a sample of 10 sweetness loss values whose mean is 0.3 or greater.

• This sample could occur quite easily by chance alone.

• This evidence against Ho is not strong. (High P-value)

Step 4: Step 4: Interpret the resultsInterpret the results in the context of in the context of the problem for our original data.the problem for our original data.

• The P-value is small,0.0006 (good evidence against null hypothesis).• meaning that we would only expect to get

this result in 6 out of 10,000 samples. • This is very unlikely, so we will reject the

null hypothesis in favor of the alternative hypothesis and conclude that the cola actually did lose sweetness.

1.02x

Statistically SignificantStatistically Significant • If the P-value is small we say that our result is

statistically significant. • The smaller the P-value, the stronger the

evidence provided by the data.How small is small enough?• Compare the P-value to the value of the

significance level α (alpha). This value is usually predetermined.

• If the P-value is as small as or smaller than α, we say that the data are statistically significant at level α.

Read pg. 527 - 532Read pg. 527 - 532