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Example Two samples of 100 electric bulbs each has a means 1500 and 1550, standard deviations are 50 and 60. Can it be concluded that two brands differ significantly in equality at 1% level of significance?

Solution: Ho : m1 - m2 i.e. there is no significant difference in the mean life of two brands of bulbs. We have = 1500, = 1500, s1 = 50, s2 = 60, n1 = 100 and n2 = 100.

Therefore, the null hypothesis Ho is rejected. Hence there is a significant difference in the mean life of the two brands of bulbs.

Example Two populations have their means equal but the standard deviation of one is twice of the other. If samples of size 2000 are drawn from each population, show that the difference between the means will almost not exceed 0.15 s' where s' is the smaller S.D.

Solution: We have s1 = s', s2 = 2s', n1 = 2000, n2 = 2000

Therefore

S =

Now

i.e.

Example The data of a certain trace element in blood is as under :
 Sample - I Sample - II Mean 28 ppm 32 ppm S. D. 14 ppm 10 ppm Size 75 male 60 female

What is the likelihood that the population means of the concentration of the elements are the same for male and female (at 95% confidence level)?

Solution: The null hypothesis is Ho : m1 - m2 and the alternative hypothesis Ha : m1 ¹ m2.

We have = 28, = 32, s1 = 14, s2 = 10, n1 = 75 and
n2 = 60.

Thus Ho is accepted i.e. the population means of the concentrations of the elements are the same for both male and female.

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Example In a certain school there are 6th graders and 7th graders. The average weight of students in a sample of 250 in 6th graders was found to be 120 lbs with s.d. of 12 lbs. While the corresponding figures in a sample of 400 students from 7th graders were 124 lbs and 141 lbs. Is the difference significant ? Also find 99% confidence limits for the difference in the average weights of students in the two graders.

Solution: We have Ho : m1 - m2 and Ha : m1 ¹ m2.

= 120, s1 = 12, n1 = 250 ® For 6th graders

=124, s2 = 14, n2 = 400 ® For 7th graders

| z | = 3.87 > 3

Hence, the difference between two samples is significant as we have rejected Ho.

Now 99% confidence limits are given by ( x1 - x2) ± 2.585

= ( 4 ) ± 2.58 ´ 1.0324

= 4 ± 2.66 which are ( 1.34, 6.66 ).

Index

8.1 Population
8.2 Sample
8.3 Parameters and Statistic
8.4 Sampling Distribution
8.5 Sampling Error
8.6 Central Limit Theorem
8.7 Critical Region
8.8 Testing of Hypothesis
8.9 Errors in Tesitng of Hypothesis
8.10 Power o a Hypothesis Test
8.11 Sampling of Variables
8.12 Sampling of Attributes
8.13 Estimation
8.14 Testing the Difference Between Means
8.15 Test for Difference Between Proportions
8.16 Two Tailed and one Tailed Tests
8.17 Test of Significance for Small Samples
8.18 Students t-distribution
8.19 Distribution of 't' for Comparison of Two Samples Means Independent Samples
8.20 Testing Difference Between Mens of Two Samples Dependent Samples or Matched Paired Observations
8.21 Chi-Square
8.22 Sampling Theory of Correlation
8.23 Sampling Theory of Regression

Chapter 1