Hypothesis Tests In statistics a hypothesis is a statement that something is true.
Hypothesis Tests In statistics a hypothesis is a statement that something is true. For example: Using a only person’s horoscope, a professional astrologer can correctly predict which of three personality charts applies to that person with probability 1/3.
Hypothesis Tests For example: Using a only person’s horoscope, a professional astrologer can correctly predict which of three personality charts applies to that person with probability 1/3. Another example: A (unbiased) jury assume a person is not guilty until proven innocent.
Hypothesis Tests The Null Hypothesis, H0, is a statement about values of a population parameter of a population. This is normally the status quo and it contains an equality. The Alternate (Research) Hypothesis, HA, is a statement that is true when the Null Hypothesis is false.
Hypothesis Tests If you make a claim that is different from the status quo, then that is the alternative hypothesis. If a 3rd party makes a claim: • If the claim contains equality, then it is the null hypothesis • If the claim does not contain equality it is the alternative hypothesis.
Hypothesis Tests In a hypothesis test: • Identify H0 and HA • Select a level of significance • Assume the null hypothesis is true • Take a sample and determine the probability of that occurring. This is called the p-value. • Reject or Fail to reject H0
Example A local brewery distributes beer in bottles labeled 341 ml. A group of college students randomly selected a sample of 50 bottles and obtained a mean of 339 ml and a standard deviation of 4.8 ml. The college students claim that the brewery is under-filling their bottles. How probable is a sample mean of 339 or lower? (Use H0: m341 over-filling HA: m<341 under-filling
Example It is highly unlikely to obtain a sample mean of 339, the fact that we obtained it gives sufficient evidence to reject the null hypotheses, that is, there is sufficient evidence to support the claim that the brewery is underfilling their bottles.
Example A poll of 100 randomly selected car owners revealed that the mean length of time that they plan to keep their cars is 7.01 years and the standard deviation is 3.74 years. The president of the Trebor Car Park is trying to plan a sales campaign targeted at car owners who are ready to buy a different car. Test the claim of the sales manager, who authoritatively states that the mean time all car owners plan to keep their cars is less than 7.5 years. Use a 0.05 significance level.
Hypothesis Tests That was a one tailed test. Examples of one tailed tests: H0: HA: OR H0: HA: In a two tailed test the alternate hypothesis contains inequality. For example: H0: HA:
Hypothesis Tests In a two tailed hypothesis test: • Identify H0 and HA • Select a level of significance • Assume the null hypothesis is true • Take a sample and determine the probability of that occurring and multiply it by two. This is the p-value. • Reject or Fail to reject H0
Example Suppose that we want to test the hypothesis with a significance level of .05 that the climate has changed since industrialization. Suppose that the mean temperature throughout history is 50 degrees. During the last n=40 years, the mean temperature has been 51 degrees with a standard deviation of 2 degrees. What can we conclude? H0: HA:
Rejection Regions Suppose that = .05. We can draw the appropriate picture and find the z score for: . We call the outside regions the rejection regions.
Rejection Regions We call the blue areas the rejection region since if the value of z falls in these regions, we reject the null hypothesis
Example 50 smokers were questioned about the number of hours they sleep each day. Test the hypothesis that the smokers need less sleep than the general public which needs an average of 7.7 hours of sleep. Compute a rejection region for a significance level of .05. If the sample mean is 7.5 and the standard deviation is .5, what can you conclude? H0: HA:
Hypothesis Tests (with rejection regions) • Identify H0 and HA • Select a level of significance • Assume the null hypothesis is true • Find the rejection region • Take a sample and determine the corresponding z-score • Reject or Fail to reject H0
Exercises • #8.8, 8.11, 8.13 on page 375 • #8.42, 8.44, 8.49, 8.52 page 389 • #8.26, 8.34 on page 381
Homework • Review Chapters 8.1-8.3 • Read Chapters 8.4, 8.5, 8.7