The following diagram shows two important c

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The following diagram shows two important c haracteristics of t-distribution. . Normal Distribution t-distribution for sample size n = 2 t-distributon for sample size n = 15 1. A t-distribution is lower at the mean and higher at the tails than a normal distribution. 2. t-distribution has proportionately greater area in its tail than the normal distribution. The t-Table The t-table gives the probability integral of t-distribution. It gives, over a range of values of v , the probabilities of exceeding by chance value of t at different levels of significance. The t-distribution has a different value for each degree of freedom and
131 when degrees of freedom are infinitely large, the t-distribution is equivalent to the normal distribution and the probabilities shown in normal distribution tables are applicable. Application of the t-distribution 1. To test significance of the mean of a random sample. To determine whether the mean of a sample drawn from a normal population deviates significantly from a stated value (the hypothetical value of the populations mean), when variance of the population is unknown we calculate the t-statistic as follows: t = n s X μ - Where x = the mean of the sample µ = the actual or hypothetical mean of the population n = the sample size s = 1 ) ( 2 - - Σ n x x = ( 29 Σ - Σ - n d d n 2 2 1 1 Where d = deviations from the assumed mean.
132 If the calculated value of |t| exceeds t α we say that the difference between x and µ is not significant and hence the sample might have been drawn from a population with mean = µ. Confidence Limits of Population Mean Assuming that the sample is a random sample from a normal population of unknown mean the 95% confidence limits of the population mean (µ) are: 05 . 0 t n s X ± And 99% limits are 01 . 0 t n s x ± Example 1: The manufacturer of a certain make of electric bulbs claims that his bulbs have a mean life of 25 months with a standard deviation of 5 months. A random sample of 6 bulbs gave the following values. Life in months 24, 26, 30, 20, 20 18 Can you regard the producers claim to be valid at 1% level of significance? Solution: Let us take the hypothesis that there is no significant difference in the mean life of bulbs in the sample and that of the population applying t-test.
133 t = n s X μ - Calculation of x and s x x- x (x- x ) 2 24 +1 1 26 +3 9 30 +7 49 20 -3 9 20 -3 9 18 -5 25 ∑x=138 ∑(x- x ) 2 = 102 x = n x Σ s = ( 1 2 - - Σ n x x = 6 138 = 1 6 102 -
134 = 23 = 5 102 = 4 . 20 = 4.517 t = 6 517 . 4 | 25 23 | - = 517 . 4 449 . 2 2 x = 1.084 v = n-1 = 6-1 = 5 For v=5, t 0.01 = 3.365 The calculated value of t is less than the table value. The hypothesis is accepted. Hence, the producer’s claim is not valid at 1% level of significance. Example 2 A random sample of size 16 has 53 as mean. The sum of the squares of the deviations taken from the mean is 135. Can this sample be regarded as taken from the population having 56 as mean? Obtain 95% and 99% confidence limits of the mean of the population.
135 Solution: H o : there is no significant difference between the sample mean and Hypothetical population mean.

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