A sample of 14 from a population produced a mean of 55.8 and a standard deviatio
ID: 3230297 • Letter: A
Question
A sample of 14 from a population produced a mean of 55.8 and a standard deviation of 7. A sample of 20 from another population produced a mean of 49.7 and a standard deviation of 10. Assume that two populations are normaly distributed and the standard deviations of the two populations are not equal.
The null hypothesis is taht the two populations means are equal ,while the alternative hypothesis is that two populations means are different. the significance level is 10% .
what is the value of the test statistic t rounded to three decimal places
Explanation / Answer
Solution:-
The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0, i.e., the two population means are equal.
Alternative hypothesis: 1 - 2 0, i.e., the two population means are not equal.
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = sqrt[(72/14) + (102/20)] = 2.91547594742
DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = (72/14 + 102/20)2 / { [ (72 / 14)2 / (13) ] + [ (102 / 20)2 / (19) ] }
DF = 72.25 / (0.942307692308 + 1.3157894736875) = 31.9959659345 or 32 (rounding to the nearest whole number)
t = [ (x1 - x2) - d ] / SE = [ (55.8 - 49.7) - 0 ] / 2.91547594742 = 2.092
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.