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Hypothesis Testing - Islamic University of · PDF file 2018. 2. 10. · Hypothesis Testing Hypothesis testing for a population mean (one sample test) A hypothesis is a statement about

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    Hypothesis Testing Hypothesis testing for a population mean (one sample test) A hypothesis is a statement about one (or more) population parameters.

    There two hypotheses

    1- Null hypothesis:

    - The null hypothesis is the statement that there is "no effect" or "no difference", that is why the word "null" is used. It is denoted by H0 and always contains the

    equality "=" sign. It should be stressed that researchers frequently put forward a

    null hypothesis in the hope that they can discredit it. The null hypothesis is always of the form

    H0: Population parameter = specified number

    2- Alternative hypothesis:

    - The alternative hypothesis in turn is the "opposite" of the null hypothesis, that is, there is an effect or difference. It is the hypothesis that we try to establish and is

    denoted by Ha. It never contains the equality "=" sign.

    The alternative hypothesis is one of the following cases

    Either null hypothesis (H0) or the alternative

    hypothesis (Ha) is true, but not both i.e. they

    cannot simultaneously be true.

    Steps of Hypothesis Testing (p-value approach)

    The p-value is defined informally as the probability of obtaining the study results by

    chance if the null hypothesis is true. When you perform a hypothesis test in statistics,

    a p-value helps you determine the significance of your results.

    1. Collect the data, i.e., obtain a random sample from the population(s) of interest. 2. Decide whether a one- or a two-tailed (sided) test is appropriate; this decision

    depends on the research question.

    Two tailed One tailed

    Left-tailed Right -tailed

    H0: µ = µ0 H0: µ = µ0 H0: µ = µ0

    Ha: µ ≠ µ0 Ha: µ < µ0 Ha: µ > µ0

    Note: µ0 is the assumed value of the population mean

    3. State a Null (H 0 ) and Alternative hypothesis (Ha).

    4. Choose a level of significance  ( = .001, .01, or .05) 5. Specify the rejection region/s.

    The location of the rejection region depends on whether the test is one-tailed or

    two-tailed.

    a. For a one-tailed test in which the symbol ">" occurs in Ha, the rejection region

    consists of area (=) in the upper tail of the sampling distribution.

    Rejection region = a

    if Ha:  > 0

    Rejection region = a

    if Ha:  > 0

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    b. For a one-tailed test in which the symbol "

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    For two tailed test, the p-value is the total of the areas to the left and the right of the

    test statistic.

    9. Check the associated probability (p-value) 10. Making decision

    √ If p-value falls within the rejection region i.e. p-value < , H0 is rejected at the predetermined  level, then we can say that the result is statistically

    significant.

    √ If p-value does not fall within the rejection region i.e. p-value ≥ , H0 is not rejected at the predetermined  level, then we can say that the result is

    statistically not significant.

    Researchers and statisticians generally agree on the following conventions for

    interpreting p-values

    p-value Result is:

    p > 0.05 not significant

    p ≤ 0.05 significant

    p ≤ 0.01 highly significant

    p ≤ 0.001 very highly significant

    One-sample test of hypothesis about a population mean

    The one sample test is a statistical procedure used to compares the mean of your

    sample data to a known value.

    Example 1 In 128 patients under 12 years of age with a particular congenital heart defect, the

    mean intensive care unit stay after surgery was 4.7 days with a standard deviation of

    7.8. Can we conclude that the average intensive care unit (ICU) stay of patients under

    12 with this defect is more than 3.5 days? Use  = 0.05

    Solution

    Note that the sample size n = 128 is sufficiently large so that the sampling distribution

    of x is approximately normal and that s provides a good approximation to . Since

    the required assumption is satisfied, we may proceed with a large-sample test of

    hypothesis about .

    1. Data: see the previous example 2. A one-sided test 3. Formulate the hypotheses as

    H0:  = 3.5

    Ha:  > 3.5

    Z0 + test statistic

    -test

    statistic

    P value is the total

    standard area

    Z0 + test statistic

    -test

    statistic

    P value is the total

    standard area

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    4. The significance level  = 0.05

    5. Specify the rejection region We are dealing with one-tailed test in which the

    symbol ">" occurs in Ha, the rejection region is an area of 0.05 in the upper tail of the

    sampling distribution of the standardized test statistic.

    Figure Rejection region for Example

    6-7. Choose and Compute the value of the test statistic,

    0 4.7 3.5 1.741 / 7.8 / 128

    x z

    s n

        

    8. Find the probability that the test statistic is in the tail beyond the calculated value

    i.e. P (z > 1.741) = 1 – P (z ≤ 1.741), the p-value for this test is 1 – 0.9591 = 0.0409. 9. Check the associated probability (p-value). This value is < .

    10. Decision Since the p-value fall within the rejection region i.e.,

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    5. The rejection region We are dealing with one-tailed test in which the symbol "

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    Figure Location of rejection region

    6-7. The value of the test statistic is computed as follows:

    61.1 180/5

    906.90

    /

    0  

     

     ns

    x z

    8. Find the probability that the test statistic is in the tail beyond the calculated value

    Since this example is dealing with two-sided test, the p-value is computed as the sum

    of the two tail areas. We go to the Z table and see what probability values are in the

    two tails beyond the points Z = -1.61 and Z = 1.61. We find that 0.0537 is in the right

    tail beyond Z = 1.61 and the same amount is in the left tail beyond Z = -1.61.

    So the p-value of the result is 0.0537 + 0.0537 = 0.1074

    9. Check the associated probability (p-value). 0.1074 is > .

    10. Decision H0 is not rejected at the predetermined  level. Consequently, the result

    is statistically not significant.

    Conclusion At α = 0.05, we can not conclude that the average vitamin A intake of 13-

    24-month-old infants is different from 90, the recommended daily allowance for

    vitamin A.

    Hypothesis tests about the difference between two population means, 1 and 2 (Comparing two means)

    The two sample test is a statistical procedure used to compares the means of two

    samples.

    Hypothesis

    ONE -TAILED TEST

    H0: 1 = 2 Ha: 1>2 or Ha: 1 < 2

    TWO -TAILED TEST

    H0: 1 = 2

    Ha: 1  2

    Test statistic: 1 2 2 2

    1 2

    1 2

    x x z

    s s

    n n

     

    Assumptions:

    1. The sample sizes n1 and n2 are sufficiently large (n1  30 and n2  30). 2. The samples are selected randomly and independent from the target populations.

    Example

    In a study on pregnant women in their third trimester who delivered during Ramadan

    or the first two weeks of Shawwal, the birthweight of the baby (in kg) was measured

    for independent random samples of babies of fasting and non-fasting women. The

    results of the investigation are summarized in the Table below. Does this evidence

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    indicate that the mean of the baby of a non-fasting mother is significantly different

    from the mean of the baby of a fasting mother? Use a significance level of  = .05.

    Non-fasting Fasting

    n1 = 75

    00.31 x

    s1 = .11

    n2 = 64

    95.22 x

    s2 = .09

    Solution 1. Data See the previous example 2. A two-sided test 3. Formulate the hypotheses The researcher wants to test the hypotheses

    H0: (1 - 2) = 0 (i.e., no difference between means of babies)

    Ha: (1 - 2) ≠ 0 (i.e., there is a difference between the mean weigh of b