SOLUTION: The amount of electricity bills for all households in a city have screwed probability distribution with a mean of $140 and a standard deviation of $30. Find the probability that th

Question 1177434: The amount of electricity bills for all households in a city have screwed probability distribution with a mean of $140 and a standard deviation of $30. Find the probability that the mean amount of electricity bills for a random sample of 75 selected from this city will be more than the population mean by at least 4.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's break down this problem step by step.
1. **Define the Random Variable**
Let X be the random variable representing the amount of electricity bills for all households in the city. We are given that X has a skewed distribution with a mean (µ) of $140 and a standard deviation (σ) of $30.
2. **Central Limit Theorem**
Since we are dealing with a sample mean (x̄) from a skewed distribution, we can apply the Central Limit Theorem (CLT). The CLT states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, as long as the sample size is large enough (typically n ≥ 30). In this case, we have a sample size of 75, which is sufficient to apply the CLT.
3. **Calculate the z-score**
We want to find the probability that the sample mean (x̄) is greater than the population mean (µ) by at least $4. This can be expressed as:
P(x̄ ≥ µ + 4) = P(x̄ ≥ 140 + 4) = P(x̄ ≥ 144)
To standardize this, we calculate the z-score:
z = (x̄ - µ) / (σ / √n) = (144 - 140) / (30 / √75) ≈ 1.15
4. **Find the Probability**
Now, we need to find the probability that the z-score is greater than 1.15. Using a standard normal table or calculator, we find:
P(Z ≥ 1.15) ≈ 0.1251
Therefore, the probability that the mean amount of electricity bills for a random sample of 75 households will be more than the population mean by at least $4 is approximately **0.1251**.
RELATED QUESTIONS
The amounts of electricity bills for all households in a particular city have an... (answered by Theo)

The amounts of electricity bills for all households in a city have a skewed probability... (answered by Theo)

For the households in Ankara, we know that the average amount of money spent on... (answered by Boreal)

According to the records of an electric company serving the Boston area, the mean... (answered by ewatrrr)

the mean monthly utility bill for a sample of households in a city is $91.79 with a... (answered by Boreal)

According to the records of an electric company serving the Boston area, the mean... (answered by Boreal)

A rainstorm in Portland, Oregon, wiped out the electricity in 7% of the households in the (answered by stanbon)

A rainstorm in Portland, Oregon, wiped out the electricity in of the households in the... (answered by stanbon)

American households increasingly rely on cell phones as their exclusive telephone... (answered by ikleyn)