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“Students” t-test. Recall: The z-test for means The Test Statistic.

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Presentation on theme: "“Students” t-test. Recall: The z-test for means The Test Statistic."— Presentation transcript:

1 “Students” t-test

2 Recall: The z-test for means The Test Statistic

3 Comments The sampling distribution of this statistic is the standard Normal distribution The replacement of  by s leaves this distribution unchanged only if the sample size n is large.

4 For small sample sizes: The sampling distribution of is called “students” t distribution with n –1 degrees of freedom

5 Properties of Student’s t distribution Similar to Standard normal distribution –Symmetric –unimodal –Centred at zero Larger spread about zero. –The reason for this is the increased variability introduced by replacing  by s. As the sample size increases (degrees of freedom increases) the t distribution approaches the standard normal distribution

6

7 t distribution standard normal distribution

8 The Situation Let x 1, x 2, x 3, …, x n denote a sample from a normal population with mean  and standard deviation . Both  and  are unknown. Let we want to test if the mean, , is equal to some given value  0.

9 The Test Statistic The sampling distribution of the test statistic is the t distribution with n-1 degrees of freedom

10 The Alternative Hypothesis H A The Critical Region t  and t  /2 are critical values under the t distribution with n – 1 degrees of freedom

11 Critical values for the t-distribution  or  /2

12 Critical values for the t-distribution are provided in tables. A link to these tables are given with today’s lecturetables

13 Look up df Look up 

14 Note: the values tabled for df = ∞ are the same values for the standard normal distribution, z  …

15 Example Let x 1, x 2, x 3, x 4, x 5, x 6 denote weight loss from a new diet for n = 6 cases. Assume that x 1, x 2, x 3, x 4, x 5, x 6 is a sample from a normal population with mean  and standard deviation . Both  and  are unknown. we want to test: versus New diet is not effective New diet is effective

16 The Test Statistic The Critical region: Reject if

17 The Data The summary statistics:

18 The Test Statistic The Critical Region (using  = 0.05) Reject if Conclusion: Accept H 0 :

19 Confidence Intervals

20 Confidence Intervals for the mean of a Normal Population, , using the Standard Normal distribution Confidence Intervals for the mean of a Normal Population, , using the t distribution

21 The Data The summary statistics:

22 Example Let x 1, x 2, x 3, x 4, x 5, x 6 denote weight loss from a new diet for n = 6 cases. The Data: The summary statistics:

23 Confidence Intervals (use  = 0.05)

24 Summary Statistical Inference

25 Estimation by Confidence Intervals

26 Confidence Interval for a Proportion

27 The sample size that will estimate p with an Error Bound B and level of confidence P = 1 –  is: where: B is the desired Error Bound z  is the  /2 critical value for the standard normal distribution p* is some preliminary estimate of p. Determination of Sample Size

28 Confidence Intervals for the mean of a Normal Population, 

29 The sample size that will estimate  with an Error Bound B and level of confidence P = 1 –  is: where: B is the desired Error Bound z  is the  /2 critical value for the standard normal distribution s* is some preliminary estimate of s. Determination of Sample Size

30 Confidence Intervals for the mean of a Normal Population, , using the t distribution

31 Hypothesis Testing An important area of statistical inference

32 To define a statistical Test we 1.Choose a statistic (called the test statistic) 2.Divide the range of possible values for the test statistic into two parts The Acceptance Region The Critical Region

33 To perform a statistical Test we 1.Collect the data. 2.Compute the value of the test statistic. 3.Make the Decision: If the value of the test statistic is in the Acceptance Region we decide to accept H 0. If the value of the test statistic is in the Critical Region we decide to reject H 0.

34 Determining the Critical Region 1.The Critical Region should consist of values of the test statistic that indicate that H A is true. (hence H 0 should be rejected). 2.The size of the Critical Region is determined so that the probability of making a type I error, , 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 1.Find the sampling distribution of the test statistic when is H 0 true. 2.Locate the Critical Region in the tails (either left or right or both) of the sampling distribution of the test statistic when is H 0 true. Whether you locate the critical region in the left tail or right tail or both tails depends on which values indicate H A is true. The tails chosen = values indicating H A.

36 3.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 H 0 true is the desired level of  =P[type I error] Sampling distribution of test statistic when H 0 is true Critical Region - Area = 

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

39 The Test Statistic

40 Critical Region (dependent on H A ) Alternative Hypothesis Critical Region

41 The z-test for the mean of a Normal population (large samples)

42 Situation A sample of n is selected from a normal population with mean  (unknown) and standard deviation . We want to test either

43 The Test Statistic

44 Critical Region (dependent on H A ) Alternative Hypothesis Critical Region

45 The t-test for the mean of a Normal population (small samples)

46 Situation A sample of n is selected from a normal population with mean  (unknown) and standard deviation  (unknown). We want to test either

47 The Test Statistic

48 Critical Region (dependent on H A ) Alternative Hypothesis Critical Region

49 Testing and Estimation of Variances

50 Let x 1, x 2, x 3, … x n, denote a sample from a Normal distribution with mean  and standard deviation  (variance  2 ) The point estimator of the variance  2 is: The point estimator of the standard deviation  is:

51 The statistic has a  2 distribution with n – 1 degrees of freedom Sampling Theory

52 Critical Points of the  2 distribution 

53 Confidence intervals for  2 and .  /2 

54 Confidence intervals for  2 and . It is true that from which we can show and

55 Hence (1 –  )100% confidence limits for  2 are: and (1 –  )100% confidence limits for  are:

56 Example In this example the subject is asked to type his computer password n = 6 times. Each time x i = time to type the password is recorded. The data are tabulated below:

57 95% confidence limits for the mean 

58 95% confidence limits for  95% confidence limits for  2

59 Testing Hypotheses for  2 and . Suppose we want to test: The test statistic: If H 0 is true the test statistic, U, has a  2 distribution with n – 1 degrees of freedom: Thus we reject H 0 if

60  /2 Accept Reject

61 One-tailed Tests for  2 and . Suppose we want to test: The test statistic: We reject H 0 if

62  Accept Reject

63 Or suppose we want to test: The test statistic: We reject H 0 if

64  Accept Reject

65 Example The current method for measuring blood alcohol content has the following properties –Measurements are 1.Normally distributed 2.Mean  = true blood alcohol content 3.standard deviation 1.2 units A 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

66 The data are tabulated below:

67 To test: The test statistic: We reject H 0 if Thus we reject H 0 if  = 0.05.

68 Two sample Tests


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