Chapter 9 - Hypothesis Testing

When the p-value is small...

...our observation would be really unlikely if the null hypothesis were true, so we should conclude the null hypothesis is false, and go with the reality presented in the alternative hypothesis instead.

When the p-value is large...

...our observations happen fairly often just by chance when the null hypothesis is correct, so we have no real evidence that it's wrong.

What are the two conditions for a hypothesis test about proportions?

1) randomization: either random sampling or random assignment (this also implies that the observations are independent) 2) sample size: the sample size needs to be sufficiently large so that the sampling distribution of the sample proportion, p-hat, is appropriately normal. The approximate normality happens when npo >(or equal to) 15, and when npo <(or equal to) 15

What are the four steps for performing hypothesis tests about proportions?

1. Establish Hypothesis - null and alternative 2. Conditions for model - normal distribution: randomization and sample size 3. Mechanics - test statistic and p-value 4. Conclusions - reject or accept

What are the four steps for hypothesis tests about means?

1. Establish hypothesis - null and alternative 2. Conditions for model - randomization and nearly normal 3. Mechanics - test statistic and p-value 4. Conclusions - reject or accept

What are the four steps for performing significance tests (hypothesis tests)?

1. Identify hypothesis 2. Check conditions for model - determines whether we use normal or T for our sampling distribution 3. Mechanics - find the test statistic and p-value 4. Make conclusions about the hypothesis

What are the two conditions for hypothesis tests about means?

1. randomization: we need our sample to be random and independent 2. Nearly Normal: we need at least one of the following to be true: a) the population distribution is approximately bell-shaped (unimodal and symmetric) b) the distribution of the data is approximately bell-shaped (recall that the data and the population tend to have similar distributions) c) the sample size is at least 30 (this is the Central Limit Theorem)

Why do we use the T-distribution instead of the normal distribution for hypothesis tests about means?

Because the standard error is calculated using a standard deviation from the sample rather than from the population.

When we are performing hypothesis tests for a population proportion, we are measuring a __________ variable.

Categorical

What formula do we use to find our degrees of freedom?

DF=n-1

What are the three formulas for the alternative hypothesis for hypothesis tests about proportion?

Ha: p<po (left-tailed test, one-tailed test) Ha: p>po (right-tailed test, one-tailed test) Ha: p/=po (two-way test, two-tailed test)

What are the formulas for the alternative hypothesis tests about means?

Ha: μ<μo (left-tailed test, one-tailed test) Ha: μ>μo (right-tailed test, one-tailed test) Ha: μ/=μo (two-sided test, two-tailed test)

What is the formula for the null hypothesis in hypothesis tests about proportions?

Ho: p=po

What is the formula for the null hypothesis tests about means?

Ho: μ=μo

When does a Type I error occur?

It occurs when we a reject a Ho that was actually true.

When does a Type II error occur?

It occurs when we do not reject Ho, but it is false.

What will the hypothesis tell us?

It will tell us the probability of obtaining the results we did if the null hypothesis were true.

What is hypothesis testing?

It's a formal statistical method of determining whether or not to reject a claim based upon i) the preponderance of evidence and ii) our level of skepticism/tolerance for error

When performing hypothesis tests about means, to find our p-value that will inform our decision, we need to use the _____________, and it's associated calculator.

T-distribution

What does the alternative hypothesis represent?

The alternative hypothesis represents some possible alternative to that which is established in the null.

What does the null hypothesis represent?

The null hypothesis tends to represent some status quo or baseline.

What is the p-value?

The probability of getting results at least as unexpected as what we observe if the null hypothesis were true. It gives some level of evidence for or against the claim presented in the alternative hypothesis.

What is the formal cut off between 'large' and 'small'?

This formal cutoff is called the significance level (a), it can change from application to application, but the default level for statisticians is a=0.05

When would use the alternative hypothesis Ha: p/=po (two-way test, two-tailed test)?

To test if the parameter is not equal to a certain value, or different from a value.

When would we use the alternative hypothesis Ha: p>po (right-tailed test, one-tailed test)?

To test if we wanted to see if the parameter is greater than some value, or if it has increased.

When would we use the alternative hypothesis Ha: p<po (left-tailed test, one-tailed test)?

To test if we wanted to test that the parameter is less than some value, or if it has decreased, or other language of the sort.

A hypothesis test begins by establishing two opposing _________.

Two opposing hypothesis: null hypothesis and the alternative hypothesis

Which type of error is this: Concluding a drug is effective when it really is not.

Type I error

What type of error is this: Saying you do not have COVID-19 when, in fact, you do.

Type II error

How to we find the p-value for hypothesis tests about proportions?

Using normal distribution. The direction you look will depend upon the alternative hypothesis.

When we make conclusions about hypothesis tests, we make them based on...

non-zero probabilities (i.e. the p-value)

Our ___________ is actually the probability of making a Type I error.

significance level, a

What formula do we use to find the test statistic for hypothesis tests about means?

t= x-bar - μ/ (s/square rt of n)

With hypothesis tests, we start off by assuming the null hypothesis is _____.

true

For hypothesis about proportions, if the p-value > a, we...

we fail to reject Ho. There is insufficient evidence, at a, Ha is true.

For hypothesis tests about proportions, if the p-value < a, we...

we reject Ho. There is sufficient evidence, at a, Ha is true.

What formula do we use to find the test statistic for hypothesis about proportions?

z = p-hat - po / square rt of (po)(qo/n)