Hypothesis Testing

PHANTOMS

parameter, hypothesis, assumptions, name of test, test statistics, obtain p-value, make decision, sate the conclusion in context

Confidence level interval

-critical value ≤ test statistic ≤ +critical value Range within which we fail to reject the null for a two-tailed test at a given level of significance

reduces power

Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Increasing the significance level reduces the region of acceptance, which makes the hypothesis test more likely to reject the null hypothesis, thus increasing the power of the test. Since, by definition, power is equal to one minus beta, the power of a test will get smaller as beta gets bigger.

What happens when sample size is doubled?

Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Thus, it increases the power of the test. The effect size is not affected by sample size. And the probability of making a Type II error gets smaller, not bigger, as sample size increases

What condition allows your to find standard error?

Independence and the 10% rule

What condition allows you to use the critical z value for hypothesis tests on proportion and confidence intervals for proportion?

Normality (large counts)

H_o: p=.5 H_a: p>.5 alpha=.1 p-value =.08 What is your conclusion?

Reject the Null. There is convincing evidence that the true population proportion is greater than .5.

Reject Hο if:

Test statistic > upper critical value or Test statistic < lower critical value

Null Hypothesis

The null hypothesis, denoted by H0, is usually the hypothesis that sample observations result purely from chance.

Significance Level

The probability of making a Type I Error and designated alpha letter

Power of a hypothesis

The probability of not committing a Type II error is called the power of a hypothesis test.

Region of Acceptance

The region of acceptance is a range of values. If the test statistic falls within the region of acceptance, the null hypothesis is not rejected. The region of acceptance is defined so that the chance of making a Type I error is equal to the significance level.

P-value

The strength of evidence in support of a null hypothesis is measured by the P-value. Suppose the test statistic is equal to S. The P-value is the probability of observing a test statistic as extreme as S, assuming the null hypotheis is true. If the P-value is less than the significance level, we reject the null hypothesis.

state conclusion

at the () significance level, there is/ is not sufficient evidence to conclude "Context of Alternative Hypothesis"

H_o: innocent H_a: guilty What is type 1 error?

found guilty but is actually innocent

How does the p value of a 1 tailed test compare to a 2 tailed using the same data?

half

For the normal condition for 1 and 2 sample confidence intervals, what do you use in your large counts inequality?

in 1 sample, use p hat (sample proportion) and sample size and for 2 sample, use each p hat (both sample porportions) and each sample size (since you need to check each sampling distribution)

test statistic

measures how far a sample statistic diverges from what e would expect if the null hypothesis were true, in standardized units

When you find the test statistic for a 2 sample z test for p1-p2, what values do you use to find the standard error of the sampling distributions for the difference of p's?

n1, n2, phat-combined (pooled p hat)

Decreasing significance level

will increase probability of failing to reject the null, and decreasing power of test. and vice versa

Fill in the blank: Factor of power (increase significance level means_______________________)

■ higher the power of the test. If you increase the significance level, you reduce the region of acceptance. As a result, you are more likely to reject the null hypothesis. This means you are less likely to accept the null hypothesis when it is false; i.e., less likely to make a Type II error. Hence, the power of the test is increased.

Fill in the blank: Factor of power (increase only in sample size means _________________________)

■greater power of the test.

Fill in the blank: Factor of power (The greater the difference between the "true" value of a parameter and the value specified in the null hypothesis,

■the greater the power of the test. That is, the greater the effect size, the greater the power of the test.

Alternative

The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

Type II error

A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called Beta, and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.

H_o: p=.57 H_a: p does not equal alpha=.05 All conditions are met. Confidence Interval is (.42,.63) What conclusion can you make?

Fail to Reject the Null at the 95% confidence level. There is not convincing evidence that the true population proportion is not .57.

Condition 1: What condition allows the sample statistic to be used in the test statistic and center of the confidence interval?

Random

H_o: p=.57 H_a: p does not equal alpha=.05 All conditions are met. Confidence Interval is (.42,.52) What conclusion can you make?

Reject the Null at the 95% confidence level. There is convincing evidence that the true population proportion is not .57.

Region of Rejection

The set of values outside the region of acceptance is called the region of rejection. If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance.

Interpretation of Confidence Interval

We are _____ % confident that the interval from _____ to _____ captures the true ________ (in context).

Construct a 95% confidence interval for finding the proportion of...

1 sample z interval for p

H_o: innocent H_a: guilty What is type 2 error?

found innocent but is actually guilty

What test would you use for these hypotheses? H_o: p=.5 H_a: p<.5

1 sample z test for p

What z-value represents a 90% CI?

1.64

What z-value represents a 95% CI?

1.96

Construct a 99% confidence interval for finding the difference in proportions of...

2 sample z interval for p1-p2

What test would you use for these hypotheses? H_o: p1=p2 H_a: p1>p2

2 sample z test for difference of proportions (p1-p2)

What z-value represents a 99% CI?

2.58

Type I error

A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level. This probability is also called alpha, and is often denoted by α.

Two-tailed test

A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test. For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10.

One-tailed test

A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution, is called a one-tailed test. For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located located on the right side of sampling distribution; that is, a set of numbers greater than 10.

H_o: p=.5 H_a: p>.5 alpha=.01 p-value =.08 What is your conclusion?

Fail to Reject the Null. There is not convincing evidence that the true population proportion is greater than .5.

For 2 sample hypothesis test for difference of proportions, why do you use the pooled p hat for standard error?

Hypothesis tests are run assuming the null hypothesis is true, which means that p1=p2...if that is the case, both sample proportions can be combined since they are assumed to come from the same distribution

Interpretation of Confidence Level

If we took many many samples of the same size from this population, about _____ % of the resulting intervals would capture the actual parameter value.