3 CommentsThe sampling distribution of this statistic is the standard Normal distributionThe replacement of s by s leaves this distribution unchanged only if the sample size n is large.
4 For small sample sizes: The sampling distribution ofis called “students” t distribution with n –1 degrees of freedom
5 Properties of Student’s t distribution Similar to Standard normal distributionSymmetricunimodalCentred at zeroLarger spread about zero.The reason for this is the increased variability introduced by replacing s by s.As the sample size increases (degrees of freedom increases) the t distribution approaches the standard normal distribution
8 The SituationLet x1, x2, x3 , … , xn denote a sample from a normal population with mean m and standard deviation s. Both m and s are unknown.Letwe want to test if the mean, m, is equal to some given value m0.
9 The Test StatisticThe sampling distribution of the test statistic is the t distribution with n-1 degrees of freedom
10 The Alternative Hypothesis HA The Critical Regionta and ta/2 are critical values under the t distribution with n – 1 degrees of freedom
11 Critical values for the t-distribution a or a/2
12 Critical values for the t-distribution are provided in tables Critical values for the t-distribution are provided in tables. A link to these tables are given with today’s lecture
14 Note: the values tabled for df = ∞ are the same values for the standard normal distribution, za …
15 ExampleLet x1, x2, x3 , x4, x5, x6 denote weight loss from a new diet for n = 6 cases.Assume that x1, x2, x3 , x4, x5, x6 is a sample from a normal population with mean m and standard deviation s. Both m and s are unknown.we want to test:New diet is not effectiveversusNew diet is effective
16 The Test StatisticThe Critical region:Reject if
27 Determination of Sample Size The sample size that will estimate p with an Error Bound B and level of confidence P = 1 – a is:where:B is the desired Error Boundza/2 is the a/2 critical value for the standard normal distributionp* is some preliminary estimate of p.
28 Confidence Intervals for the mean of a Normal Population, m
29 Determination of Sample Size The sample size that will estimate m with an Error Bound B and level of confidence P = 1 – a is:where:B is the desired Error Boundza/2 is the a/2 critical value for the standard normal distributions* is some preliminary estimate of s.
30 Confidence Intervals for the mean of a Normal Population, m, using the t distribution
31 An important area of statistical inference Hypothesis TestingAn important area of statistical inference
32 To define a statistical Test we Choose a statistic (called the test statistic)Divide the range of possible values for the test statistic into two partsThe Acceptance RegionThe Critical Region
33 To perform a statistical Test we Collect the data.Compute the value of the test statistic.Make the Decision:If the value of the test statistic is in the Acceptance Region we decide to accept H0 .If the value of the test statistic is in the Critical Region we decide to reject H0 .
34 Determining the Critical Region The Critical Region should consist of values of the test statistic that indicate that HA is true. (hence H0 should be rejected).The size of the Critical Region is determined so that the probability of making a type I error, a, is at some pre-determined level. (usually 0.05 or 0.01). This value is called the significance level of the test.Significance level = P[test makes type I error]
35 To find the Critical Region Find the sampling distribution of the test statistic when is H0 true.Locate the Critical Region in the tails (either left or right or both) of the sampling distribution of the test statistic when is H0 true.Whether you locate the critical region in the left tail or right tail or both tails depends on which values indicate HA is true.The tails chosen = values indicating HA.
36 the size of the Critical Region is chosen so that the area over the critical region and under the sampling distribution of the test statistic when is H0 true is the desired level of a =P[type I error]Sampling distribution of test statistic when H0 is trueCritical Region - Area = a
37 The z-test for Proportions Testing the probability of success in a binomial experiment
38 Situation A success-failure experiment has been repeated n times The probability of success p is unknown. We want to test either
50 Let x1, x2, x3, … xn, denote a sample from a Normal distribution with mean m and standard deviation s (variance s2)The point estimator of the variance s2 is:The point estimator of the standard deviation s is:
51 Sampling Theory The statistic has a c2 distribution with n – 1 degrees of freedom
65 ExampleThe current method for measuring blood alcohol content has the following propertiesMeasurements areNormally distributedMean m = true blood alcohol contentstandard deviation 1.2 unitsA new method is proposed that has the first two properties and it is believed that the measurements will have a smaller standard deviation.We want to collect data to test this hypothesis.The experiment will be to collect n = 10 observations on a case were the true blood alcohol content is 6.0