Hypothesis Tests Hypothesis Tests Large Sample 1- Proportion z-test

What are hypothesis tests? Calculations that tell us if a value occurs by random chance or not – if it is statistically significant Is it... –a random occurrence due to variation? –a biased occurrence due to some other reason?

Nature of hypothesis tests - First begin by supposing the “effect” is NOT present Next, see if data provides evidence against the supposition Example:murder trial How does a murder trial work? First - assume that the person is innocent must Then – must have sufficient evidence to prove guilty Hmmmmm … Hypothesis tests use the same process!

Steps: 1)Define the parameter 2)Hypothesis statements 3)Assumptions 4)Calculations (Find the p-value) 5)Conclusion, in context Notice the steps are the same except we add hypothesis statements – which you will learn today

Writing Hypothesis statements: Null hypothesis – is the statement being tested; this is a statement of “no effect” or “no difference” Alternative hypothesis – is the statement that we suspect is true H0:H0: Ha:Ha:

The form: Null hypothesis H 0 : parameter = hypothesized value Alternative hypothesis H a : parameter > hypothesized value H a : parameter < hypothesized value H a : parameter = hypothesized value Hypotheses ALWAYS refer to populations (parameters)

For each pair of hypotheses, indicate which are not legitimate & explain why Must use parameter (population) not a statistic (sample) Must use same number as H 0 ! H 0 MUST be “=“ ! Must be NOT equal!

Writing Hypotheses H 0 : p = 0.3; H a : p < 0.3 : p = 0.5; : p ≠ 0.5 H0H0 HaHa H0H0 : p = 0.2; : p > 0.2HaHa

Steps: 1)Define the parameter 2)Hypothesis statements 3)Assumptions 4)Calculations (Find the p-value) 5)Conclusion, in context

Assumptions SRS from population Success/Failure Condition (Large enough sample) np  and n(1-p)   rule – the sample is less then 10% of the population YEA YEA – These are the same assumptions as confidence intervals!!

Steps: 1)Define the parameter 2)Hypothesis statements 3)Assumptions 4)Calculations (Find the p-value) 5)Conclusion, in context

P-values - The probability that the test statistic would have a value as extreme or more than what is actually observed In other words... is it far out in the tails of the distribution?

Level of significance - Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true Is the probability that we will reject the null hypothesis, assuming that it is true Denoted by  –Can be any value –Usual values: 0.1, 0.05, 0.01 –Most common is 0.05 (default value)

Statistically significant – as smallsmallerThe p-value is as small or smaller than the level of significance (  ) fail to rejectIf p > , “fail to reject” the null hypothesis at the  level. rejectIf p < , “reject” the null hypothesis at the  level.

Facts about p-values: ALWAYS make decision about the null hypothesis! Large p-values show support for the null hypothesis, but never that it is true! Small p-values show support that the null is not true. Double the p-value for two-tail (=) tests Never acceptNever accept the null hypothesis!

Never “accept” the null hypothesis!

At an  level of.05, would you reject or fail to reject H 0 for the given p-values? a).03 b).15 c).45 d).023 Reject Fail to reject

Formula for hypothesis test:

Calculating p-values One sided test (<) P-value = P (z < calculated value) One sided test (>) P-value = P (z > calculated value) Two sided test (  ) P-value = 2P (z < calculated value)

Steps: 1)Define the parameter 2)Hypothesis statements 3)Assumptions 4)Calculations (Find the p-value) 5)Conclusion, in context

Writing Conclusions: Decision: A statement of the decision being made (reject or fail to reject H 0 ) & why (linkage) Conclusion: A statement of the results in context. (state in terms of H a ) AND

A statement about H a in context (words)! Decision: Conclusion: There is enough evidence to conclude that the true proportion of... Since the p-value > , I fail to reject the null hypothesis at the  level. Since the p-value  , I reject the null hypothesis at the  level. or There is not enough evidence to conclude that the true proportion of...

A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?

Parameter and Hypotheses H 0 : p =.2 H a : p >.2 p = the true proportion of people who heard the ad 1)The sample must be random which is stated in the problem. 2)The sample should be large. Since np = 400(.2) = 80 >10 and n(1-p) = 400(.8) = 320 >10, the sample is large enough. 3)The sample must be less then 10% of the population. The population should be at least 4000 people which I will assume it to be. Use the parameter in the null hypothesis to check assumptions! Assumptions (Conditions) Since the conditions are met, a z-test for proportions is appropriate.

Calculations

Since the p-value > , I fail to reject the null hypothesis at the.05 level. Conclusion: Decision: There is not enough evidence to conclude that the true proportion of people who heard the ad is greater than.2