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Tests of Small Samples

 

(a) Testing of A Single Mean. Here sample is small (n < 30) and cr is unknown. 1. Set up H0 : µ = µ0 2. Set up H1 : µ > µ0 or µ < µ or µ ≠ µ0 3. Test statistic

distribution with (n - 1) degrees of freedom.

4. Set up level of significance a and the critical value as !tab (from table of t - distribution).

5. Compute the statistic say tcal

6. Decisions

All ttab based on (n - I) degrees of freedom.

 

Example 5: the mean breaking strength of a certain kind of metallic rope is 160pounds. If six pieces of ropes (randomly selected from different rolls) have a mean breaking strength of 154.3 pounds with a standard deviation of 6.4 pounds, test the null hypothesis 1..1. = 160 pounds against the alternative hypothesis 1..1. < 160 pounds at 1% level of significance. Assume that the population follows normal distribution.

 

 

 

Solution. 1. H0 : µ= 160 pounds

 

2. H1: µ< 160 pounds

 

3. Since n = 6, the test statistic is taken as

Example 5. The mean breaking strength of a certain kind of metallic rope is 160 pounds. If six pieces of ropes (randomly selected from different rolls) have a mean breaking strength of 154.3 pounds with a standard deviation of 6.4 pounds, test the null hypothesis 1-L = 160 pounds against the alternative hypothesis 1-L < 160 pounds at 1% level of significance. Assume that the population follows normal distribution.

 

Solution. 1. H0 : µ = 160 pounds

2. H1 : µ < 160 pounds

3. Since n = 6, the test statistic is taken as

4. α = O.OI, H1 indicates it is left tailed test. Critical value at 6- 1 i.e., 5 degrees of              freedom is -3.365

 

5. Computation

6.   Decision

Since        tcal > -3.365

 

  •  it lies in the acceptance region
  •  H0 is accepted
  •  Mean breaking strength of the metallic rope can be taken as I60 pounds.

 

(b) Testing of Difference of Two Means. Here nl,n2 or both are small (< 30) and the population variances are unknown but equal and two populations follow normal distribution.

 

1.   Set up H0  : µ1- µ2  = k

2.   Set up H1  : µ1- µ2  > k,     or µ1- µ2  <  k  or  µ1- µ2 ≠ k

3.   Test statistic :

where ,

Here, the statistic follows t - distribution with n1 + n2- 2 degrees of freedom.

4.  Set up the level of significance, a. and the critical value say t13b at n1 + n2 - 2 degrees of     freedom.

5.  Compute the test statistic as tcal·

6. Decision

Example 6. The following are the number of sales with a sample of 6 sales people of gas lighter in a city A and a sample of 8 sales people of gas lighter in another city B made over a certain fixed period of time :

City A :     63,   48,   54,      44,    59,   52

City B :    41,   52,    3850,   66,   54,   44,   61

 

 

Assuming that the populations sampled can be approximated closely with normal distributions having the same variance, test H0 : µ1 = µ2 against H1 :  µ1 ≠ µ2 at the 5% level of significance.

 

Solution.    1. H0 : µ1 = µ2

2. H1 :  µ1 ≠ µ2

 

3. Test statistic:

where ,