Hypothesis testing, P-value, β-error, sample size for β<=0.1

[s3a]

Homework Statement

The life in hours of a battery is known to be normally distributed, with standard deviation of 1.5 hours.
A random sample of 12 batteries has a sample mean life time of 50 hours.

a) Test the hypothesis that the mean battery life is 50.5 hours (by using α = 0.05).

b) What is the P-value for the test in part (a)?

c) Find the β-error for the test in part (a).

d) What sample size is required to ensure that beta does not exceed 0.10, if the true mean life is 52 hours?

The solution is included as TheSolution.jpg.

Homework Equations

z = (Xbar - μ)/( σ √(N) ), where Xbar is sample mean, σ is the population standard deviation and N is the sample size.

The Attempt at a Solution

I'm struggling a lot with this topic, and the fact that the solutions aren't very descriptive makes it harder to learn.

I'm confused for even basic stuff. For example, What's the final answer to part (a)? Is the null hypothesis rejected because 87.49% < 95%? Also, I'm confused about one or two alpha tails, and I have no idea if this problem has one or two of them, or if it's just not related to this topic. Basically, how do I know the accuracy rate is 95% and not 90%?

Also, for part (b), I don't see why 2*(1 - 0.8749) is being computed.

I don't see anything about part (c). Where is the β value?

As for part (d), could someone explain what the solution is doing?

Any input would be GREATLY appreciated!

 

[FactChecker]

You can not reject the hypothesis that the mean is 50.5 at the 95% confidence level because the odds of the 12-sample mean of a true mean=50.5 being as far or farther than 0.5 away is 2*(1-0.8749) = 0.2502. That is much > 0.05. It is 2 tail because the sample could be either higher or lower than 50.5 for you to reject the hypothesis that mean=50.5. Use one sided if you are trying to prove higher but not lower, or trying to prove lower but not higher.

 

[s3a]

Okay the part about the amount of tails is making a little more sense to me, but I still don't get the rest.

For example, the solution says "Since .25 > .05[,] the battery life is not 50.5 hours." Doesn't that mean that the null hypothesis should be rejected?

Put differently, the probability that the null hypothesis is false (i.e. the battery life being 50.5 hours) should be at most 5%, but if I understand correctly, that probability is roughly 25%, which means that the odds of the null hypothesis not being valid are greater than the margin of error that was chosen.

So, are you sure that the hypothesis can not be rejected?

 

[Stephen Tashi]

Put differently, the probability that the null hypothesis is false (i.e. the battery life being 50.5 hours) should be at most 5%,

Perhaps you meant to say "the probability the null hypothesis is rejected" instead of "the probability the null hypothesis is false".

This problem does not involve computing the probability that the null hypothesis is false ( or the probability that it is true). It involves assuming the null hypothesis is true and computing the probability of an event defined in terms of the observed data. The probabilities P(A given B) and P(B given A) are, in general, not the same number. The calculations are done with the assumption "given the null hypothesis is true". You aren't computing the probability that the null hypothesis is true "given the observed data".

 

[FactChecker]

I disagree with some things in the solution attachment. The larger the probability of the tails, the harder it is to prove anything, including rejecting the null hypothesis. So the fact that the tail probabilities add up to 0.25 means that you can not conclude anything at the 95% confidence level. A second objection I have is that it says that the null hypothesis is that the mean is 50. That is the sample mean. The hypothesis should be made before a sample is taken and should not be formed based on the same sample that is used to draw a conclusion. That practice can distort the probabilities.