Question

Questions from : Fundamentals of Biostatistics by Bernard Rosner

Suppose we wish to test the hypothesis H0: µ = 45 vs. H1:  µ > 45.

7.23 What will be the result if we conclude that the mean is  greater than 45 when the actual mean is 45? (i)  We have made a type I error. (ii)  We have made a type II error. (iii)  We have made the correct decision. 7.24 What will be the result if we conclude that the mean is  45 when the actual mean is 50? (i)  We have made a type I error. (ii)  We have made a type II error. (iii)  We have made the correct decision. Suppose we wish to test H0: µ = 30 vs. H1: µ ≠ 30 based on  a sample of size 31.  7.25 Which of the following sample results yields the smallest p-value and why? (i)  x = 28, s = 6 (ii)  x = 27, s = 4 (iii)  x = 32, s = 2 (iv)  x = 26, s = 9

Answer 1

7.23)

It is given that the true mean is 45. That means the null hypotheiss is true. But we wrongly reject the null hypothesis and accept the alternative. That means we have made an error. The following are the 2 types of errors that we can makre

• Type I error occurs when we reject a null hypothesis that is true
• Type II error occurs when we accept a null hypothesis that is false

In this case we have rejected the null hypothesis when it actually true.

Ans: (i)  We have made a type I error.

7.24) In this case it is given that the true mean is 50, that means the null hypothesis is wrong. But we wrongly accept the null hypothesis and say that true mean is 45. This is an error and we have committed Type II error as per the definition given above.

Ans: (ii)  We have made a type II error.

7.25)

We also know that the sample size n=31

The smallest p-value will be for the sample which will give the largest absolute z value. The absolute value of z is calculated using

This fraction will be the largest for the sample with smallest standard error of mean . Since n=31 is the same for all the samples. the standard error of mean will be the least for a sample with smallest standard deviation s. From the options we can see that option iii has the smallest standard deviation s=2. Hence it is most likely to have the largest absolute z value and hence the smallest p-value

ans: (iii)  x = 32, s = 2

The actual calculations are shown below

The folowing are the p-values for each of the options

i) The sample mean and sample standard deviation s=6 and the standard error of mean

The z value is given by

p value is P(Z<-1.85) = 0.032

ii)

The sample mean and sample standard deviation s=4 and the standard error of mean

The z value is given by

p value is P(Z<-4.17) = 0.000 (using excel formula =NORM.S.DIST(-4.17,TRUE))

iii)

The sample mean and sample standard deviation s=2 and the standard error of mean

The z value is given by

p value is P(Z> 5.56) = 0.000 (using excel formula =NORM.S.DIST(5.56,TRUE))

iv)

The sample mean and sample standard deviation s=9 and the standard error of mea

The z value is given by

p value is P(Z<-2.47) = 0.007 (using excel formula =NORM.S.DIST(-4.17,TRUE))