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“Students” t-test

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Recall: The z-test for means The Test Statistic

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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.

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

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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

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t distribution standard normal distribution

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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.

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

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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

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Critical values for the t-distribution or /2

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

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Look up df Look up

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

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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

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The Test Statistic The Critical region: Reject if

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The Data The summary statistics:

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

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Confidence Intervals

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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

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The Data The summary statistics:

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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:

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Confidence Intervals (use = 0.05)

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Summary Statistical Inference

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Estimation by Confidence Intervals

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Confidence Interval for a Proportion

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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

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Confidence Intervals for the mean of a Normal Population,

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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

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

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Hypothesis Testing An important area of statistical inference

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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

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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.

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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]

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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.

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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 =

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The z-test for Proportions Testing the probability of success in a binomial experiment

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Situation A success-failure experiment has been repeated n times The probability of success p is unknown. We want to test either

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The Test Statistic

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

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

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

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The Test Statistic

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

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

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

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The Test Statistic

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

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Testing and Estimation of Variances

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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:

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

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Critical Points of the 2 distribution

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Confidence intervals for 2 and . /2

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

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

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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:

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95% confidence limits for the mean

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95% confidence limits for 95% confidence limits for 2

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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

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/2 Accept Reject

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

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Accept Reject

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

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Accept Reject

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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

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The data are tabulated below:

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

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Two sample Tests

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