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HYPOTHESIS TESTING

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Purpose The purpose of hypothesis testing is to help the researcher or administrator in reaching a decision concerning a population by examining a sample from that population The purpose of hypothesis testing is to help the researcher or administrator in reaching a decision concerning a population by examining a sample from that population

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Hypothesis It is a statement about one or more population It is a statement about one or more population It is usually concerned with the parameter of the population about which the statement is made It is usually concerned with the parameter of the population about which the statement is made

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Research Hypothesis It is the assumption that motivate the research. It is usually the result of long observation by the researcher. This hypothesis led directly to the second type of hypothesis It is the assumption that motivate the research. It is usually the result of long observation by the researcher. This hypothesis led directly to the second type of hypothesis

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Statistical Hypothesis This is stated in a way that can be evaluated by appropriate statistical technique. This is stated in a way that can be evaluated by appropriate statistical technique.

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Statistical hypothesis It is composed of two types: It is composed of two types: Null hypothesis( Ho): It is the particular hypothesis under test, and it is the hypothesis of no difference Null hypothesis( Ho): It is the particular hypothesis under test, and it is the hypothesis of no difference Alternative hypothesis (H A ): which disagree with the null hypothesis Alternative hypothesis (H A ): which disagree with the null hypothesis

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Test Statistic It is a mathematical expression of sample values which provides a basis for testing a statistical hypothesis. It is a mathematical expression of sample values which provides a basis for testing a statistical hypothesis. The result of this test will determine whether we will accept the null hypothesis and so the H A will be rejected, or we reject the null hypothesis and so the H A will be accepted. The result of this test will determine whether we will accept the null hypothesis and so the H A will be rejected, or we reject the null hypothesis and so the H A will be accepted.

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Errors There are two possible errors to come to the wrong conclusion: There are two possible errors to come to the wrong conclusion: Type 1 error: rejection of the null hypothesis when it is true. It is presented by alpha, which is the level of significance, often the 5%, 1%, and 0.1% (α=0.05, 0.01, and 0.001) levels are chosen. The selection depends on the particular problem Type 1 error: rejection of the null hypothesis when it is true. It is presented by alpha, which is the level of significance, often the 5%, 1%, and 0.1% (α=0.05, 0.01, and 0.001) levels are chosen. The selection depends on the particular problem

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P-value It is the smallest value of α for which the Ho can be rejected, so it gives a more precise statement about probability of rejection of Ho when it is true than the alpha level, so instead of saying the test statistic is significant or not, we will mention the exact probability of rejecting the Ho when it is true It is the smallest value of α for which the Ho can be rejected, so it gives a more precise statement about probability of rejection of Ho when it is true than the alpha level, so instead of saying the test statistic is significant or not, we will mention the exact probability of rejecting the Ho when it is true

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Steps in conducting hypothesis testing Hypothesis testing can be presented as NINE steps: Hypothesis testing can be presented as NINE steps:

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1.Data The nature of the data whether it consists of counts, or measurement will determine the test statistic to be used The nature of the data whether it consists of counts, or measurement will determine the test statistic to be used

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2.Hypotheses Null Hypothesis (Ho): which is the hypothesis of no difference, and the alternative hypothesis( H A ) Null Hypothesis (Ho): which is the hypothesis of no difference, and the alternative hypothesis( H A ) If we reject the Ho we will say that the data to be tested does not provide sufficient evidence to cause rejection. If it is rejected we say that the data are not compatible with Ho and support the alternative hypothesis (H A ) If we reject the Ho we will say that the data to be tested does not provide sufficient evidence to cause rejection. If it is rejected we say that the data are not compatible with Ho and support the alternative hypothesis (H A )

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3.Test statistic It uses the data of the sample to reach to a decision to reject or to accept the null hypothesis. The general formula for a test statistic is: It uses the data of the sample to reach to a decision to reject or to accept the null hypothesis. The general formula for a test statistic is: relevant statistic-hypothesized parameter relevant statistic-hypothesized parameter Test statistic = ----------------------------------------- standard error of the relevant statistic standard error of the relevant statistic

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4.Test statistic--Example _ x - µ x - µ Z=------------- Z=------------- σ n σ n

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5. Distribution of test statistic It is the key for statistical inference It is the key for statistical inference

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6. Decision Rule It will tell us to reject the null hypothesis if the test statistic falls in the rejection area, and to accept the it if it falls in the acceptance region It will tell us to reject the null hypothesis if the test statistic falls in the rejection area, and to accept the it if it falls in the acceptance region

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6. Decision Rule The critical values that discriminate between acceptance and rejection regions depends on alpha level of significance The critical values that discriminate between acceptance and rejection regions depends on alpha level of significance If the value of the test statistic falls in the rejection region area, it is considered statistically significant If the value of the test statistic falls in the rejection region area, it is considered statistically significant If it falls in the acceptance area it is considered not statistically significant If it falls in the acceptance area it is considered not statistically significant

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6. Decision Rule Whenever we reject a null hypothesis, there is always a possibility of type 1 error( rejection of Ho when it is true). This is why we should decrease this error to the least possible. Whenever we reject a null hypothesis, there is always a possibility of type 1 error( rejection of Ho when it is true). This is why we should decrease this error to the least possible.

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Critical values The values of the test statistic that separate the rejection region from the acceptance region The values of the test statistic that separate the rejection region from the acceptance region

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Acceptance region A set of values of the test statistic leading to acceptance of the null hypothesis A set of values of the test statistic leading to acceptance of the null hypothesis ( values of the test statistic not included in the critical region) ( values of the test statistic not included in the critical region)

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Rejection region A set of values of the test statistic leading to rejection of the null hypothesis A set of values of the test statistic leading to rejection of the null hypothesis

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7. Computed test statistic This should be computed and compared with the acceptance and rejection regions This should be computed and compared with the acceptance and rejection regions

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8. Statistical decision It consists of rejecting or not rejecting the Ho. It is rejected if the computed value of the test statistic falls in the rejection area, and it is not rejected if the computed value of the test statistic falls in the acceptance region It consists of rejecting or not rejecting the Ho. It is rejected if the computed value of the test statistic falls in the rejection area, and it is not rejected if the computed value of the test statistic falls in the acceptance region

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9. Conclusion If Ho is rejected, we conclude that H A is true. If Ho is not rejected we conclude that Ho may be true. If Ho is rejected, we conclude that H A is true. If Ho is not rejected we conclude that Ho may be true.

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Two sided test If the rejection area is divided into the two tails the test is called two-sided test, If the rejection area is divided into the two tails the test is called two-sided test,

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One sided test If the rejection region is only in one tail it is called one-sided test If the rejection region is only in one tail it is called one-sided test The decision will depend on the nature of the research question being asked by the researcher The decision will depend on the nature of the research question being asked by the researcher

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Single population mean, known population variance _ x - µ x - µ Z=------------- Z=------------- σ n σ n

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Single population mean with unknown population variance _ x - µ x - µ t =------------- t =------------- s n s n

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Difference between two populations means with known variances _ _ _ _ (X 1 – X 2 ) – ( µ 1 - µ 2 ) (X 1 – X 2 ) – ( µ 1 - µ 2 ) Z=------------------------------- Z=------------------------------- σ 2 1 /n 1 + σ 2 1 /n 2 σ 2 1 /n 1 + σ 2 1 /n 2

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Difference between two population mean with unknown and unequal variances _ _ _ _ (X 1 – X 2 ) – ( µ 1 - µ 2 ) (X 1 – X 2 ) – ( µ 1 - µ 2 ) t =------------------------------- t =------------------------------- s 2 1 /n 1 + s 2 1 /n 2 s 2 1 /n 1 + s 2 1 /n 2

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Difference between two population mean with unknown but assumed equal variances _ _ _ _ (X 1 – X 2 ) – ( µ 1 - µ 2 ) (X 1 – X 2 ) – ( µ 1 - µ 2 ) t =------------------------------- t =------------------------------- Sp 1 /n 1 + 1 /n 2 Sp 1 /n 1 + 1 /n 2

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Paired t-test _ d - µ d d - µ d t =------------- t =------------- S d n S d n

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Single population proportion ˜ P -P P -P Z=------------- Z=------------- P(1-P)n P(1-P)n

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Difference between two population proportions ˜ ˜ ˜ ˜ (P 1 -P 2 ) – (P 1 -P 2 ) (P 1 -P 2 ) – (P 1 -P 2 ) Z=----------------------------------------- Z=----------------------------------------- P 1 (1-P 1 )/n 1 + P 2 (1-P 2 )/n 2 P 1 (1-P 1 )/n 1 + P 2 (1-P 2 )/n 2

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Example A certain breed of rats shows a mean weight gain of 65 gm, during the first 3 months of life. 16 of these rats were fed a new diet from birth until age of 3 months. The mean was 60.75 gm. If the population variance is 10 gm, is there a reason to believe at the 5% level of significance that the new diet causes a change in the average amount of weight gained A certain breed of rats shows a mean weight gain of 65 gm, during the first 3 months of life. 16 of these rats were fed a new diet from birth until age of 3 months. The mean was 60.75 gm. If the population variance is 10 gm, is there a reason to believe at the 5% level of significance that the new diet causes a change in the average amount of weight gained

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Answer Ho=65 Ho=65 H A 65 H A 65 Z 1-α/2 α=0.05 Z=1.96 (critical value) Z 1-α/2 α=0.05 Z=1.96 (critical value) _ x - µ 60.75-65 x - µ 60.75-65 Z=---------- = ----------- = -5.38 Z=---------- = ----------- = -5.38 σ n 10/ 16 σ n 10/ 16 Sine the calculated values falls in the rejection region, we reject the Ho, and accept the H A

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In the above example, if the population variance is unknown, and the sample Sd is 3.84 In the above example, if the population variance is unknown, and the sample Sd is 3.84

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Answer t 1- α/2 =± 2.1315 df =n-1 df =n-1 _ x - µ 60.75-65 x - µ 60.75-65 t =-------------=------------= - 4.1315 t =-------------=------------= - 4.1315 s n 3.84/ 16 s n 3.84/ 16 Sine the calculated values falls in the rejection region, we reject the Ho, and accept the H A

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