Chapter 9: Testing a Claim

What is the goal of a confidence interval?

Confidence intervals are used to estimate a population parameter using sample data

what is a hypothesis test?

significance tests used when one wants to assess whether the data provides enough evidence of some claim about the population

Ho: P=x

• null hypothesis

• population parameter and it is always = sign

• this is the parameter we assume to be true

• the claim we weigh evidence against

Ha: P <, >, ≠, x

• alternative hypothesis

• what we think is true

• the claim we are trying to find evidence for

If the alternative hypothesis is a “less than” < or a “greater than” >,

it is called a one-sided hypothesis test or a one-tailed hypothesis test

If the alternative hypothesis is a “not equal”≠,

it is called a two-sided hypothesis test or a two-tailed hypothesis test

Hypothesis test always refer to…

the population, not a sample. Be sure to state Ho and Ha in terms of population parameters like p or µ.

Steps of a hypothesis test

Step 1: Stating the Hypotheses

• Ho: Ha: and where p is the true proportion of _________

Step 2: Conditions

• random, normality, 10% condition

Step 3: Calculations

• varies on the type of hypothesis test

Step 4: Conclusion

• first sentence stating facts (p-value and alpha level)

• second sentence writing the conclusion on the context of the problem

P-Value

the probability of getting evidence for the alternative hypothesis Ha as strong or stronger than the observed evidence when the null hypothesis Ho is true.

Small P-Values

Give convincing evidence for the alternative hypothesis Ha because it is saying that the observed result is unlikely to occur when the null hypothesis Ho is true

Large P-Values

fail to give convincing evidence for Ha because they say that the observed result is likely to occur by chance alone when Ho is true.

If the observed result is unlikely to occur by chance alone when Ho is true (small P-value)

we will “reject Ho” conclude there is convincing evidence for Ha

If the observed result is not unlikely to occur by chance alone when Ho is true (large P-Value)

we will fail to reject Ho conclude there is not convincing evidence for Ha

What is the significance level?

The value that we use as a boundary for deciding whether an observed result is unlikely to happen by chance alone when the null hypothesis is true

you compare your P-values to your alpha level

Most common alpha levels

.01, .05, .10

Type I Error

occurs if a test rejects Ho when Ho is true. That is, the test finds convincing evidence that Ha is true when it really isn’t

Type II Error

occurs if a test fails to reject Ho when Ha is true. that is, the test does not find convincing evidence that Ha is true when it really is.

T-Test

used to calculate the p-value a tstat on calculator for a mean hypothesis problem

1-proportion z test

used to calculate the p-value a zstat on a calculator for a proportion hypothesis problem

Need to write degrees of freedom on step 3 of a hypothesis test for mean

degrees of freedom (n-1)

For a combined confidence interval and hypothesis test the alpha level is determined by

1-(confidence level)

Power

the probability that the test will find convincing evidence for Ha when a specific alternative value of the parameter is true.

Equations for Power

Power= 1-P(Type II Error) and P(Type II Error) =1- Power

The power of a significance test to detect an alternative value of the parameter when Ho is false and Ha is true, based on a random sample size n and signficance level alpha, will be larger when:

• sample size n is larger

• significance level is larger (which decreases B)

• null and alternative parameter values are farther apart

probability that a type I error is committed is

alpha level ⍺

probability that a type II error is committed is

beta level ß (we don’t calculate this; it is always given)

⍺ and ß are inversely related

⍺⬆ ß ⬇

⍺⬇ ß ⬆