Chapter 10, Section 10.2, Problem 09 Rural and urban students are to be compared

Question

Chapter 10, Section 10.2, Problem 09 Rural and urban students are to be compared on the basis of their scores on a nationwide musical aptitude test. Two random samples of sizes 90 and 100 are selected from rural and urban seventh-grade students. The summary statistics from the test scores are Rural Urban Sample size90 100 76.1 82.3 Standard deviation 8.2 7.8 Mean Construct a test to determine if there is a significant difference between the population mean scores. Use = 0.05 You need to test the hypotheses Ho:M1-2- versus H1:1-2 with = 0.05 There a significant difference between the population mean scores.

Explanation / Answer

Answers

H0: µ1 - µ2 = 0 Vs H0: µ1 - µ2 0. ANSWER 1

There is a significant difference between population mean scores. . ANSWER 2

Details

Let X = score of rural seventh grade students in musical aptitude test and Y = = score of urban seventh grade students in musical aptitude test.

Then, we assume X ~ N(µ1, 12) and Y ~ N(µ2, 22), where 12 = 22 = 2, say and 2 is unknown.

Claim:

Population mean score is the same for both rural and urban students.

Hypotheses:

Null: H0: µ1 = µ2 Vs Alternative: HA: µ1 µ2

Test Statistic:

t = (Xbar - Ybar)/[s{(1/n1) + (1/n2)}] where

s2 = {(n1 – 1)s12 + (n2 – 1)s22}/(n1 + n2 – 2);

Xbar and Ybar are sample averages and s1, s2 are sample standard deviations based on n1 observations on X and n2 observations on Y respectively.

Calculations

Given Xbar = 76.1, Ybar = 82.3; s1 = 8.2, s2 = 7.8; n1 = 90, n2 = 100

s2 = (89s12 + 99s22)/188

    = (7.99)2

t = - 5.33

Decision:

Since |tcal| (5.33) > tcrit (1.973, i.e., upper2.5% point of t188), H0 is rejected.

Alternatively, p-value [2.67E-07, i.e., P(|t188| > 5.33)] is much less than 0.05 (given level of significance), H0 is rejected.

=> there is enough evidence to suggest that there is significant difference between population mean scores of rural and urban seventh grade students.

DONE

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