The lifetime of a certain smart phone battery has an

unknown distribution with mean value of 8 hours and
standard deviation of 2 hours. What is the approximate
probability that the average battery lifetime of a sample
of 36 batteries will exceed 8.1 hours?

To solve this problem, we can use the Central Limit Theorem (CLT) and the properties of the normal distribution. The Central Limit Theorem states that for a large sample size, the distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution.

Here's how we can calculate the approximate probability that the average battery lifetime of a sample of 36 batteries will exceed 8.1 hours:

1. Calculate the standard deviation of the sample mean (also known as the standard error). Since the standard deviation of the population is 2 hours and we have a sample size of 36, we can use the formula:

Standard Error = Standard Deviation / Square Root of Sample Size
= 2 / square root(36)
= 2 / 6
= 1/3

2. Calculate the z-score, which measures how many standard deviations an observation is from the mean. In this case, we want to find the probability that the average battery lifetime exceeds 8.1 hours, so we need to calculate the z-score for 8.1:

z-score = (x - mean) / standard error
= (8.1 - 8) / (1/3)
= (0.1) / (1/3)
= 0.3

3. Look up the cumulative probability in the standard normal distribution table for the calculated z-score (0.3) to find the probability that a standard normal random variable is less than 0.3. Since we want the probability that the average battery lifetime exceeds 8.1 hours, we can subtract this cumulative probability from 1:

Probability = 1 - cumulative probability for z-score (0.3)

You can use a standard normal distribution table or a statistical software package to find the cumulative probability for a z-score of 0.3 and subtract it from 1 to get the final probability.