Final answer:
The probability that the mean utility bill is greater than $105 for a sample of 16 utility bills is calculated using the Central Limit Theorem. First, find the standard error, then convert $105 to a z-score, and finally use the standard normal distribution to find the corresponding probability.
Step-by-step explanation:
To find the probability that the mean utility bill for a sample of 16 utility bills is greater than $105, we can use the Central Limit Theorem which tells us that the distribution of sample means will be normally distributed with its own mean (the same as the population mean) and its own standard deviation (the population standard deviation divided by the square root of the sample size). In this case, the mean of the sample means is $100, and the standard deviation of the sample means, often called the standard error, is $10 / √16 = $2.50.
To find the probability in question, we convert $105 to a z-score. The z-score formula is Z = (X - μ) / σ, where X is the sample mean, μ is the population mean, and σ is the standard error. Substituting our values, we get Z = ($105 - $100) / $2.5 = 2. This Z=2 tells us how many standard errors the value $105 is above the mean.
Now, we look up this Z-score on the standard normal distribution table or use a calculator that provides probabilities for the standard normal distribution. The table or calculator gives us the probability of a Z-score being less than 2. Since we want greater than $105, we subtract this value from 1 to get the probability of the mean being greater than $105.