Question

When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution with µ = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded.

a. State appropriate hypotheses for the quality control supervisor to test.

b. The supervisor wants to conduct a test using samples of only n=4 so as to avoid wasting batteries. Explain briefly why a sample size of 30 would be necessary.

c. Suppose you draw a random sample of 30 batteries and find that the mean battery life for the sample is 6.8 hours and the standard deviation of the sample is 0.6 hours. Conduct a hypothesis test at the 0.05 level to determine if there is sufficient evidence to conclude that the mean lifetime of all batteries is less than 7 hours.

d. Describe a Type I and a Type II error in this situation and the consequences of each.

Answer 1

Answer:

a)

Null hypothesis

Ho : µ = 7 ;

Alternative hypothesis

Ha : µ < 7

b)

A great example is one which really speaks to the charateristics of the populace. On the off chance that one choses lesser number of things from a populace, at that point there is a likelihood that they may veer off from the qualities of populace. This may prompt wrong outcomes.

Also in the above case, if the chief chooses just 4 batteries, there are conceivable outcomes of Type I and II blunders.

A sample size of 30 is important to ensure that the appropriation follows an ordinary dispersion.

c)

Null theory : The mean lifetime of the considerable number of batteries is under 7 hours.

Alternative hypothesis : The mean lifetime of the considerable number of batteries is more noteworthy than 7 hours.

Given,

sample n = 30

mean µ = 6.8 hours

standard deviation σ = 0.6 hours

consider,

t = (x - μ) / σ

substitute values

= (6.8 - 7)/0.6

t = - 0.33

Presently on the off chance that we check the estimation of t at 5% hugeness level, at that point we get t = 1.313, which is more noteworthy than the above determined estimation of t (0.33).

Subsequently we acknowledge the invalid theory for example the mean lifetime of the considerable number of batteries is under 7 hours.

d)

Type I mistake :

The boss is persuaded that the mean lifetime of the batteries is under 7 hours and disposes of them however the procedure was working truly fine.

Type II mistake :

The boss gets persuaded that the procedure is fine yet the mean lifetime of the batteries is under 7 hours.

The outcome of the Type I blunder would be the loss of cash since the great batteries are being tossed out and the outcome of the Type II mistake would bring about the exchange of awful batteries which would bring about client insatisfaction and it will demolish the name of the organization.