Test of hypothesis (t)

Statistical Test of Hypothesis ( Part 2)

Small-Sample Tests of Hypothesis

For cases having a small sample size a specialized ( derived) distribution is employed to provide corrections in error due to a small sample size. This distribution is called a "t - distribution"

Assumptions. test statistics

1. Sampled population is normal

2. Small random sample (n < 30)

3. σ is unknown

Properties of the t Distribution:

(i) It has n − 1 degrees of freedom (df)

(ii) Like the normal distribution it has a symmetric mound-shaped probability distribution

(iii) More variable (flat) than the normal distribution

(iv) The distribution depends on the degrees of freedom. Moreover, as n becomes larger, t converges to Z.

(v) Critical values (tail probabilities) are obtained from the t table

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t distribution tableUsing the t distribution table, critical values are determined given the form of the alternative hypothesis the level of confidence, and the degree of freedom

Example:

Find the critical values of t given that: a) Ha: ≠ , α = 0.05 df =12 t(two tailed,α,df,) = +/- 2.179

b) Ha: > , α = 0.01 df =29 t(one tailed,α,df,) = + 2.462

Small-Sample Tests of Hypothesis

Three cases for discussion

Small-Sample Inferences About a Population Mean

Small-Sample Inferences About the Difference Between Two Means: Independent Samples

Small-Sample Inferences About the Difference Between TwoMeans: Paired


Example:

The efficacy of a new medication is claimed to last for 10 hours. A random sample of 12 patients were tested and showed a mean of 9.5 hours with a standard deviation of 0.2 hours. Using 5% level of significance, test the hypothesis that the population mean is less than 10 hours.

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test statistics

Solution:

Step1 : H0 : μ = 10 (μ is not less than 10)

Ha : μ< 10 (μ is less than 10)

Step2 : df= (n-1) = (12-1)= 11, t (one tailed, 5%, 11) = -1.796

as can be seen in the decision model to the right, the t distribution "mimics" the normal distribution, if the computed value fall within the shaded area, the hull hypothesis is rejected. The rejection region is positioned on the left tail since the alternative hypothesis is denoted as "less than ( < )"


Step3 : t test since n ≤ 30

Step4: Decision: Reject Ho, Accept Ha

Conclusion: Th efficacy of a new medication is less than 10 hours.

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Review Excercises

Case Study:

A panel survey was conducted to by the National Population Commission to study the general effects of aging on the sociological and physiological well being of senior citizens living in selected urban areas in Metro Manila. 24 senior citizens in Barangay Nagkaisa participated in the study. The life expectancy of the participants were measured at 78.23 years with sd of 7.2 years.

If the national life expectancy is 72 years is there sufficient evidence that the respondents lived longer? Level of significance is at 5%.

Small-Sample Inferences About the Difference Between Two Means:

Independent Samples


Assumptions.1. Normal populations2. Small samples ( n1 < 30; n2 < 30)3. Samples are randomly selected4. Samples are independent5. Variances are equal with common variance

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test statistics

Note: Sp is the pooled estimate of variance which means that the test statistics assumes that s1 and s2 are treated as equal. Likewise df = n1+n2-2.


Example: A training is administered to two groups of nurses. The first group (n1= 15) were trained using the lecture/demo method

while the second group (n2=11) used programmed materials and video. At the end of the course, group 1 trainees were rated at 86 with a standard deviation of 3 while the second group were rated at an average of 84 with a standard deviation of 4. Is there sufficient evidence that the lecture/demo method is better? Let α = 1%

Note: before solving ensure that all the necessary data is laid out to minimize errors

rating

Method n (sample size) mean sd

lecture/ demo 15 86 3.0

programmed materials and video.

11 84 4.0

Solution: Step1 : H0 : μ1 = μ2 (there is no significant difference in the two methods)

Ha : μ1 > μ2 (the lecture/demo method is better)

Step2 : df = 24, t (one tailed, 1%, 24) = + 2.492


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Step3 : t test ( pooled estimate of variance)

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Step4: Decision: Accept Ho

Conclusion:There is no significant difference between the two methods. The lecture/ demo method is not better than programmed instructions.

Review Excercises

Refer to the results in the case study "magnets and pain relief"

pain before (Score_1)

pain after (Score_2)

n (sample size)

mean sd mean sd

Active Magnet

29 9.6206 0.7277 4.3793 3.1443

Inactive Placebo

21 9.5238 0.8728 8.4285 1.8593

Test the hypothesis (alpha= 1%):

a) The two groups has comparable pain levels before the "magnets" were used.

b) Bioflex magnets were more effective in pain management

Small-Sample Inferences About the Difference Between Two Means:

Paired

Where,

average difference of the paired values

hypothesized difference

individual difference of the paired values

sample size

standard deviation

Small-Sample Inferences About the Difference Between Two Means: Paired

Assumptions.1. Normal populations2. Small samples ( n1 < 30; n2 < 30)3. Samples are randomly selected4. Samples are not independent (paired)

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Example: An intervention program is administered to seven ( 7 ) patients who experienced myocardial

infarction or cardiac surgery. The program involves coaching to improve self efficacy of the participants to perform a range of activities frequently encountered in daily living ( bathing, walking, dressing up etc). The design involves 2 weeks of coaching. Self efficacy ratings were taken before and after the coaching activities.

The results are illustrated in the table below:

Before After

1 98 106

2 105 110

3 117 113

4 132 116

5 136 117

6 114 119

7 159 124

Test the hypothesis that there is a significant difference in the self efficacy ratings at 5% level of significance.


Solution: Step1 : H0 : μ1 = μ2 (there is no significant difference in the efficacy levels)

Ha : μ1 ≠ μ2 (there is significant difference in the efficacy levels)

Step2 : t (two tailed, 5%, 6 ) = +/- 2.447

Before After d d2

1 98 106 -8 64

2 105 110 -5 25

3 117 113 4 16

4 132 116 16 256

5 136 117 19 361

6 114 119 -5 25

7 159 124 35 1225

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Step3 : t test ( paired)

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Step4: Decision: Accept Ho t= 1.33

Conclusion:There is no significant difference before and after the intervention. The intervention was not effective in improving efficacy levels of the patients

Review ExcercisesCase study :

A medical transcription firm has observed errors in the transcription of documents forwarded by their clients to their company. As part of their effort in improving product/service quality they instituted changes in the workplace environment. Test hypothesis at 5% level of significance that the improvement in the work place environment has an effect in reducing transcription errors. Data is presented in the table

Before After

1 8.70 8.20

2 6.60 6.30

3 7.00 6.90

4 7.60 7.10

5 8.40 8.40

6 9.00 8.50

7 9.40 9.20

8 9.60 9.60

9 11.30 10.50

10 12.30 10.90

End of

Second Part of the Discussions

Test of hypothesis (t)

Health & Medicine

Transcript of Test of hypothesis (t)