Exercise 1.1 One hundred students were given an Algebra test. A random sample of
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Exercise 1.1 One hundred students were given an Algebra test. A random sample of ten students was taken out of class of 800 enrolled students. The time it took each student to complete the test is recorded below. 22.2, 23.7, 16.8, 18-3, 19.7, 16.9, 17-2, 18.5, 21.0, and 19.7 a. Find the mean, variance and standard deviation for this sample of ten students b. Construct a 95% confidence interval for the population meantime to complete this Algebra test c. Test if the population mean time to complete the test is 22.5 minutes. Exercise 1.2 Asurvey often households showed the following number of TV sets per household. 1, 3,2, 1, 1, 2,0, 2, 1, 2 a. Find the mean, variance and standard deviation for this sample of ten households b. Construct a 95% confidence interval for the population mean number of TV sets per household. c. Test if the population mean umber of TV sets per household is 2.Explanation / Answer
Solution – Time for Algebra Test
Back-up Theory
Let Xi = time taken by the ith student, i = 1,2, , 10 …………………………. (1)
Sample mean: Mean X = (sum Xi over 1 to n)/n where n = sample size …… (2)
Sample variance: s2 = {sum (Xi – Mean X)2 over 1 to n)/(n – 1) ………………(3)
Sample standard deviation; s = sq.rt(s2) ………………………………………..(4)
100(1- )% Confidence Interval for Population Mean µ = {Mean X ± (s.tn – 1,/2)/n} …. (5)
where tn – 1,/2 is the upper /2 percentage point of t-distribution with degrees of freedom = n – 1
To test the null hypothesis H0: µ = µ0 vs alternative HA: µ µ0,
Test statistics is t = n|Mean X - µ0|/s …………………………………………. (6)
t has a t-distribution with degrees of freedom = n – 1 …………………………..(7)
Decision criterion: Reject H0, if calculated value tcal > ttab …………………….(8)
where tcal is the calculated value of test statistics and ttab = upper /2 percentage point of t-distribution with degrees of freedom = n – 1.
Now, to work out the problem,
Given, n = 10, Mean X = 19.4; s2 = 5.4378; s = 2.37 ANSWER 1
Given = 5%, 95% Confidence Interval for Population Mean µ = 19.4 ± (2.37x2.262)/10
= 19.4 ± (5.3609)/3.1622 = 19.4 ± 1.70 ANSWER 2
Test /statistics: t = 3.1622|(19.4 – 22.5)|/2.37 = 3.1622x3.1/2.37 = 4.136.
Since 4,136 > 2.262, H0 is rejected. ANSWER 3
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