Question

Consider two samples: one which has 3 observations with a mean of 8.3 and a standard deviation of 1.2, and another which has 10 observations with a mean of 3.0 and a standard deviation of 5.4. Based on these samples, what is likely to be true for the populations if we test at the 5% level?

• A. Only equal variances
• B. Only equal means
• C. Both equal means and equal variances
• D. Neither equal means nor equal variances

Answer 1

For testing of mean,

Null hypothesis (Ho) :

Alternative hypothesis (H1) :

Test statistic is given by -

  

Where,

.

m1= first sample mean = 8.3

m2 = second sample mean = 3

n1= sample size of first sample = 3

n2= sample size of second sample = 10

s1= standard deviation of first sample = 1.2

s2 = standard deviation of second sample = 5.4

So, s =√(263.88/10) = 5.137

Therefore,

= 1.568

Critical value of t at 0.05 level of significance and 10 degrees of freedom is 2.228 (This can be obtained from t table)

Since, t calculated < t critical, hence, we may not reject the null hypothesis. Hence, the two means are equal.

For testing of Variance,

Null hypothesis:

Alternative facts:

Test statistic is given by-

f = larger sample variance/ smaller sample variance

=

=

= 20.25

Critical value of f at 9 and 2 degrees of freedom at 5% Level of significance is 19.38 as obtained from the f table.

Since, f calculated > f critical, we may reject the null hypothesis. So, variances are not equal.

So,the correct the option is (b) only equal means.