Question

Suppose that we wish to test the null hypothesis Ho that the proportion p of ledger sheets with errors is equal to .05 versus the alternative Ha , that the proportion is larger than .05, by using the following scheme. Two ledger sheets are selected at random. If both are error free, we reject Ho. If one or more contains an error, we look at a third sheet. If the third sheet is error free, we reject Ho. In all other cases, we do not reject Ho. (a) Explain type I error in the context of this problem. (b) What is the value of a associated with this test? Is this a good, sensible test? (c) Explain your answer based on the rejection rule (i.e. region) and the hypotheses. (d) Explain type II error in the context of this problem. (e) Calculate B(pa) = P(Type II error\p = pa) as a function of pa. What would be B(0.7)?

Answer 1

solution:

given data:

a)

Type I error happens when a null hypothesis is true and we reject it. The probability of type I error is a . In the given question we reject the null hypothesis based on the previously mentioned plot.

b)

In the previously mentioned question the null hypothesis  is dismissed when

Case 1: Both the randomly selected ledgers are error free
Case 2: When in any event one of them is without mistake and we draw a third record and it is found to be error free. In the Case 1, the likelihood of a record being error free is 0.5

The likelihood that both the ledgers are error free is 0.25

The Case 2 can be separated into two sections

(1) When both the ledgers drawn are mistaken

For this situation the likelihood of dismissing the null hypothesis  is 0.125

(2) At the point when one of the two drawn register is error blunder.

For this situation the likelihood of rejecting null hypothesis is 0.125

The likelihood of committing the type 1 error is 0.25+0.125+0.125 = 0.50

So the estimation of $\alpha$ is 0.5.

The test can't be considered a a reasonable one since it has no measurable basis.

c)

The district of rejection has no statistical basis. The hypothesis  is rejected  based on the irregular drawing of records . The test measurement doesn't keep any standard appropriation and can't be viewed as a reasonable one.

d)

The type 2 error is committeed when the alternative hypothesis is accepted even when it is false

the probability of type 2 error is

$\beta$ = 1 - $\alpha$

$\beta$ = 0.50

e)

here

The acceptance of alternate hypothesis even when it is wrong implies that p = 0.05

$\beta$ = p/10

$\beta$(0.7) = 0.07

Therefore,

$\beta$(0.7) = 0.07

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