18. = 13. I. One class, of 20 students, averages 75 on a test with variance, s A

Question

18. = 13. I. One class, of 20 students, averages 75 on a test with variance, s Another class, of 21 students, averages 78 on the same test, with (a) Compute the estimated standard error of the difference of sample means. (You have to identify exactly what this number is, here.) (b) Are the averages significantly different at the 5% level? Include calculation of the test statistic and p-value. Use Tl test for df. Report the df the TI gives. Also find the critical value. State Ho and H1 2. With the same classes as in problem 4., decide if the sample variance of the first class is significantly higher than that of the second class, at the 5% level. Include the calculation of the State Ho and H1. Assume normal test statistic and p-value. Also find the critical value. distributions

Explanation / Answer

Here we have to test the hypothesis that,

H0 : mu1 = mu2 Vs H1 : m1 not= mu2

where mu1 and mu2 are two population means.

Assume alpha = level of significance = 5% = 0.05

Given that,

s1 and s2 are two standard deviations.

Standard deviation is the square root of variances.

Now we have given sample data and sample sizes are less than 30 so we use two-sample t test.

By assuming equal variances we use two sample t-test.

a) Estimated standard error of the difference of sample means.

Standard error = Sp* sqrt(1/n1 + 1/n2)

where Sp is the pooled standard deviation.

Sp = sqrt { [(n1-1)s1^2 + (n2-1)s2^2] / n1+n2-2}

Sp = sqrt { [(20-1)*18 + (21-1)*13 ] /20+20-2} = 3.98

SE = Sp * sqrt [1/n1 + 1/n2]

= 3.98* sqrt [1/20 + 1/21] = 1.24

b) Now we have to find test statistic.

t = (X1bar - X2bar) / SE

= (75-78) / 1.24 = -2.42

Now we have to find p-value for taking the decision.

P-value we can find in EXCEL.

syntax :

=TDIST(x, deg_freedom, tails)

where x is absolute value of test statistic.

deg_freedom = n1+n2-2 = 20+21-2 = 39

tails = 2

P-value = 0.0203

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : There is sufficient evidence to say that two population means are differ.

2) Now we have to test the hypothesis that,

H0 : Two variances are equal

H1 : First variance is greator than second one.

Assume alpha = level of significance = 5% = 0.05

Test statistic follows F-distribution.

The test statistic is,

F = largest variance / smallest variance

= 18/13 = 1.38

Now we have to find p-value.

p-value in EXCEL :

syntax :

=FDIST(x, deg_freedom1, deg_freedom2)

x is test statistic

deg_freedom1 = n1-1 = 20-1 = 19

deg_freedom2 = n2-1 = 21-1 = 20

P-value = 0.2381

P-value > alpha

Accept H0 at 5% level of significance.

Conclusion : Two variances are equal.

n1 20 n2 21 x1 75 x2 78 s1^2 18 s2^2 13 s1 4.24 s2 3.61

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