Question

In a clinic, a random sample of 60 patients is obtained, and each person’s red blood cell
count (in cells per microliter) is measured. The sample mean is 5.28. The population
standard deviation for red blood cell counts is 0.54.

(a) At the 0.01 level of significance, test the claim that the sample is from a population

with a mean less than 5.4. [4 marks]
(b) What is the Type 11 error ,6 in the hypothesis testing of Part (a) if the true mean of
the population is 5.17? [7 marks]

(0) The Type 11 error fl in Part (b) is considered to be too large and it is decided to
reduce fl to 0.1. With the same sample size, find the new Type I error a. [7 marks]

Answer 1

Here number of patients = 60

sample mean = 5.28

population standard deviation = 0.54

(a) Here at 0.01 level of significance,

standard error of sample mean = 0.54/sqrt(n) = 0.0697

Z = (5.28 - 5.40)/0.0697 = -1.7213

p - value = Pr(Z < -1.7213) = 0.0426

so here p - value is greater than 0.01 so populaion mean is not less than 5.4.

(b) Now, here we will reject the null hypothesis if

< 5.4 - Z_0.01 se₀

< 5.4 - 2.3263 * 0.0697

< 5.2378

Pr(Type II error) = Pr( > 5.2378 ; 5.17; 0.0697)

Z = (5.2378 - 5.17)/0.0697 = 0.9727

Pr(Type II error) = 1 - Pr( < 5.2378 ; 5.17; 0.0697) = 1 - Pr(Z < 0.9727) = 1 - 0.8346 = 0.1654

(c) Now probabbility of type II error is 0.1 and it is decided to reduce it to 0.1

so, requisite sample mean is

Pr(x > ; 5.17 ; 0.0697) = 0.1

Z = 1.2816

( - 5.17)/0.0697 = 1.2816

= 5.17 + 0.0687 * 1.2816 = 5.258

Now

we have to find type I error = Pr( < 5.258 ; 5.4 ; 0.0697)

Z = (5.258 - 5.4)/0.0697 = -2.0373

New Type I error = Pr(Z < -2.0373) = 0.02

so new type I error is 0.02.