Question

Activity One: Hypothesis Testing: Impact of Sample Size A marketing claim made by Buzz Electronics is that the average life of a battery they provide to various devices exceeds 4000 hours. To test this claim, a random sample of just 12 components is examined, tracing the life between installation and failure. The data revealed an average life of 4366 hours with a sample standard deviation of 1001 hours. (Adapted from Groebner, p. 325) (i) What conclusion should be reached based on the sample data assuming a 0.05 level of significance? (ii) Suppose that the random sample consisted of 500 components. What is your conclusion now? (iii) What assumptions regarding normality need to be made in both (i) and (ii)?

Answer 1

(i) The Hypotheses are:

at 0.05 level of significance

Rejection region:

Reject Ho if T(obs) >t_0.05,11=1.796, here t- distribution and t test is applicable since n<30.

Test statistic:

where, n= sample size

S= standard deviation,

P- value :

P- value associated with T value is calculated by table shown below hence,

0.1<P-value<0.25

Conclusion:

Since T(obs) =1.267 <t(0.05,11) =1.796 and the P-value is greater than 0.05 then we fail to reject and conclude that at the 5% level of significance there is insufficient evidence to support the claim of a true mean exceeds 4000.

(ii) Now if sample size is increased to 500 then Z test will be applicable since n>30. so,

The Hypotheses are:

at 0.05 level of significance

Rejection region:

Reject Z(obs)>Z_0.05=1.645

and P- value would be almost 0.

Conclusion:

Since Z(obs) =8.176>>Z(0.05)=1.645 and P-value<<0.05, hence we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that true mean exceeds 4000.

(iii) Assumption made for normality:

1. Sample is taken from a large population so that population is normality distributed.

2. when plotted on graph , it results in a normal distribution, bell-shaped distribution curve.

3. Sample is randomly selected.