4. A random sample of size n = 10 is taken from a large population. Let µ be the

Question

4. A random sample of size n = 10 is taken from a large population. Let µ be the unknown population mean. A test is planned of H0 : µ = 12 vs. HA : µ 6= 12 using = 0.1. A QQ plot indicates it it is reasonable to assume a normal population. From the sample, ¯x = 14.2 and s = 4.88. (I suggest doing this problem with a calculator and table as practice for exams. You may check your answers with R if you wish.) (a) Since the data leave it plausible that the population is normal, and the population standard deviation is unknown, a t-test is appropriate. Compute the p-value of the test. Do you reject or not reject H0? (b) Using s = 4.88 as our best guess of , compute the power of the test if the true population mean is µA = 15. (c) Using s = 4.88 as our best guess of , approximately what sample size would be required to achieve a power of 0.8 if the true population mean is µA = 15? Give your answer as the smallest whole number that meets the criterion.

Explanation / Answer

Random sample size n = 10

H0 : µ = 12

Ha : µ ? 12

(a) Here the population standard deviation ? is unknown, a t-test is appropriate.

¯x = 14.2 and s = 4.88

Test statistic :

standard error of the mean = s/ sqrt(n) = 4.88/sqrt(10) = 1.5432

t = (¯x - µH )/(s/ sqrt(n) = (14.2 - 12)/ (4.88/sqrt(10) = 2.2 / 1.5432 = 1.426

p - value = Pr(t > 1.426) = 0.1877 < 0.05

We don't reject H0 .

(b) s = 4.88 as our best guess of ?

True population mean µA = 15

So we shall not reject the null hypothesis if Pr(¯x < ¯x0 ; 12; 1.5432) = 0.05 [ 0.05 for two sided test]

THe z value for the given p is = 1.645

Z = (¯x0 -12)/ 1.5432

1.645 = (¯x0 -12)/ 1.5432

¯x0 = 12 + 1.5432 * 1.645 = 14.5386

Here true mean = 15 so we shall not reject the null hypothesis if ¯x < 14.5386

Pr(Type II error) = Pr(¯x < 14.5386; 15; 1.5432) = ?

Z = (14.5386 - 15)/ 1.5432 = -0.30

Pr(¯x < 14.5386; 15; 1.5432) = Pr(Z < -0.30) = 0.3821

Power of the test = 1 - 0.3821 = 0.6179

(c) Here power to be achieved = 0.8

sample size let sa n

so we shall not reject the null hypothesis if ¯x < 12 + 1.645 * 4.88/ ?n

¯x < 12 + 8.0276 /?n

as power means truely detecting false null hypothesis.

standard error of the sample mean = 4.88/?n

Pr( ¯x < 12 + 8.0276 /?n ; 15; 4.88/?n) = 1 - 0.80 = 0.20

Z value for the given p - value is -0.84

-0.84 = (12 + 8.0276 /?n - 15)/ 4.88/?n

8.0276 /?n -3 = -4.099/ ?n

3 = 12.1268/ ?n

?n = 4.042266

n = 16.34 or 17

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