Large sample CI for μ Small sample CI for μ Large sample CI for p Review of CI’s Large sample CI for μ Small sample CI for μ Large sample CI for p

Large sample confidence intervals for the sample mean The Confidence Interval is expressed as: E is called the margin of error. For samples of size > 30,

Sample Size The sample size needed to estimate m so as to be (1-a)*100 % confident that the sample mean does not differ from m more than E is: …round up

Confidence Interval for the mean when s is unknown and n is small The (1- a)*100% confidence interval for the population mean m is The margin of error E, is in this case N.B. The sample must be assumed to be a random sample AND the population must be approximately normal. population.

Confidence intervals for a population proportion If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion. The idea here is to take a sample of size n, and count how many items in the sample have this attribute, call it x. Calculate the sample proportion, . We would like to use the sample proportion as an estimate for the population proportion.

FACT: The confidence interval for p is: where p = x/n and q = 1 – p. ASSUMPTION: The sample size is sufficiently large that both np and nq are at least 15.

Hypothesis Tests In statistics a hypothesis is a statement that a researcher believes is true.

Hypothesis Tests In statistics a hypothesis is a statement that something is true.

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 normally contains an equality. The Alternate (Research) Hypothesis, HA, is a statement that is true when the Null Hypothesis is false.

Hypothesis Tests Identify H0 and HA Select a level of significance Sketch the rejection region. Assume the null hypothesis is true Take a sample and compute the z-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.  

Test the hypothesis Compute z using the null hypotheses μ =50, the sample standard deviation s = 2, and the observed mean of 51 The z-value IS in rejection region, therefore can reject the null hypothesis The data does support the alternative hypothesis . Can conclude with confidence 95% that the mean temperature has changed.

Test the hypothesis The z-value IS in rejection region, therefore can reject the null hypothesis The data does support the alternative hypothesis . Can conclude with confidence 95% that the mean temperature has changed. Of course, the z-value we observed could occur even if the null hypothosis were true, but this is not likely (probability only 0.05). Thus, we could be in error in supporting that the mean temperature has changed,. This sort of error is called a Type 1 error.

Type I errors We note that we could be in error since the z-value 3.16 is unlikely but not impossible. Rejecting the null hypotheses when it is in fact true is called a Type I error. In our example, the probability of a Type I error is less than 0.05.

P-values In the previous example, would have rejected H0 even if a smaller value of alpha had been used. The smallest such value is called the p-value of the test. In the example, the normal table can be used to show p = .0002 The p-value also gives the probability of a Type I error

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