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