Probability Standard deviation

Question

NeverReady batteries has engineered a newer, longer lasting AAA battery. The company claims this battery has an average life span of 18 hours with a standard deviation of 0.5 hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span is 17.7 hours. If the process is working properly, what is the probability of getting a random sample of 30 batteries in which the sample mean lifetime is 17.7 hours or less? Is the company's claim reasonable? (Round your probability to four decimal places.)

The company's claim  ____?_______ is is not reasonable because the probability that a sample of 30 batteries would have a lifetime of at most 17.7 hours is about _____________?___________ .

Answer 1

We have that μ = 18, σ = 0.5, n = 30, x̄ = 17.7 (population mean, population standard deviation, sample size and sample mean respectively).

It is asked that the probability of getting a random sample of 30 batteries in which the sample mean lifetime is 17.7 hours or less. This is a sampling distribution problem.

We can use software to get the answer but manually this is the process:

1. You have to find the standard deviation of x̄ which is: σ / √n = 0.5 / √30 ≈ 0.091
2. Then you have to standardize: z = (x̄ - μ)/(σ / √n) = (17.7 - 18) / 0.091 = -3.29
3. Using a left-tail z table you look for the z value which will give you P (z <= -3.29) = 0.0005

If this process works properly, then the probability that a sample of 30 batteries would have at most 17.7 hours is 0.05%. Therefore, the class was justified to question the claim.