2.2.2. The sign test: Normal approximation (Large samples)

As a consequence of the central limit theorem, the normal distribution can be used to approximate the binomial distribution if the sample size is large Experts differ on the exact definition of “large”. We suggest that the normal approximation is acceptable if the sample size exceeds 20. With large n, the binomial distribution with is close to the normal distribution with mean and standard deviation .

The test statistic is

where

S-is the number of positive signs,

n-is the number of nonzero sample observations.

The null hypothesis to be tested is that the proportion p- of nonzero observations in the population that are positive is 0.5; that is

1. To test either null hypothesis

against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or

Example:

In a TV commercial, filmed live, 100 persons tested two brands of coffee: say brand A and brand B and each selected their favorite. 56 persons preferred coffee of brand A, 40 preferred coffee of brand B, and four expressed on preference. Test at the 5% significance level the null hypothesis that for the population, there is no overall preference for coffee of brand A over the brand B.

Solution:

From the information given above we obtain that

;

To test if there is no overall preference in this population for one brand of coffee over the other, the hypotheses are

(People have no preference for either brand of coffee)

(People have preference for one brand of coffee)

The decision rule is

Reject if or

The value of the test statistic is

; and .

Since is not less than we fail to reject the null hypothesis. And at 5% significance level we accept that there is no preference for either brand of coffee.

From the standard normal distribution it follows that the approximate

or 10.32%.

Hence the null hypothesis can be rejected at all significance levels greater than 10.32%.

Exercises

1. Two computer specialists estimated the amount of computer memory (in gigabytes) required by five different offices.

Office Specialist A Specialist B

1 4.2 6.1

2 6.3 6.7

3 3.1 3.0

4 2.2 2.9

5 7.2 10.9

Use the Sign test to test the null hypothesis that two specialists estimations are the same against the alternative that specialist B estimates higher than specialist A.

2. In a test of two chocolate chip cookie recipes, 13 out of 18 subjects favored recipe A. Using the sign test, find the significance probability when states that recipe A is preferable.

3. A firm attempting to determine if a difference exists in two manufacturing methods. A sample of 10 workers was selected, and each worker completed the production task using each of the two production methods.

Worker Method 1(minutes) Method 2(minutes)

1 10.2 9.5

2 9.6 9.8

3 9.2 8.8

4 10.6 10.1

5 9.9 10.3

6 10.2 9.3

7 10.6 10.5

8 10.0 10.0

9 10.7 10.2

10 10.9 10.2

Use the sign test and perform the null hypothesis that there is no overall preference for one method over the other.

4. A social researcher interviews 25 newly married couples. Each husband and wife are independently asked the question: “How many children would you like to have?” The following data are obtained

Answer of Answer of

Couple Husband Wife Couple Husband Wife

1 3 2 14 2 1

2 2 2 15 3 2

3 2 1 16 2 2

4 2 3 17 0 0

5 5 1 18 1 2

6 0 1 19 2 1

7 0 2 20 3 2

8 1 3 21 4 3

9 2 2 22 3 1

10 3 1 23 0 0

11 4 2 24 2 3

12 1 2 25 2 2

13 3 3

Use the Sign test with to test against two sided alternative the null hypothesis that, for the population of families no difference in opinions between husbands and wives.

5. A random sample of 80 sale managers was asked to predict whether next year’s sale would be higher than, lower than, or about the same as in the current year. The results are shown below. Test the null hypothesis that the opinion of managers is evenly divided on the question against a two sided alternative.

Prediction Number

Higher 37

Lower 28

About the same 15

6. Of a random sample of 120 university students, 67 expected to achieve a better GPA than last year, 48 expected a lower GPA than last year, and

5 expected about the same GPA. Do these data present strong evidence that, for population of students they are divided evenly on the expectations, against the alternative that more expect a lower GPA compared with last year?

7. Of a random sample of 150 university instructors, 62 believed that student’s skills in solving problems increased over the last decade, 54 believed these skills had deteriorated and 4 saw no change. Evaluate the strength of the sample evidence suggesting that, for all university instructors, teachers are divided evenly on the issue against the alternative that more teachers believe that student’s skills in solving problems have improved.

8. In a coffee taste test 48 individuals stated a preference for one of two well-known brands. Results showed 28 favoring brand A, 16 favoring brand B, and 4 undecided. Use the sign test with to test the null hypothesis that there is no difference in the preferences for the two brands of coffee against a two sided alternative.

Answers

1. p- value = 0.1874 or 18.74%;2. p-value = 0.0482 or 4.82%;

3. p– value = 0.1798 or 17.98%; 4. ; accept ; 5. ;

p-value = 0.06%; 6. ;p-value = 3.84%;7. ;

p-value = 22.96%; 8. ; reject .