Question 1193276: The distribution of weekly salaries at a large company is right-skewed with a mean of RM1000 and a standard deviation of RM350. What is the probability in estimating the mean weekly salary for all employees of the company will be
a) Between RM900 and RM1100?
b) Below RM900?
c) Greater than RM850?
d) Exactly RM1000?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Understand the Central Limit Theorem**
* Even though the population of weekly salaries is right-skewed, the Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases.
* **Key Assumptions:**
* The sample size is sufficiently large (often considered 30 or more).
* The samples are drawn independently and randomly from the population.
**2. Calculate the Standard Error of the Mean**
* Standard Error (SE) = σ / √n
* Where σ is the population standard deviation
* And n is the sample size
**3. Calculate Z-scores**
* To find probabilities related to the sample mean, we need to standardize the values using the z-score formula:
* z = (x̄ - μ) / SE
* Where x̄ is the sample mean
* μ is the population mean
* SE is the standard error
**4. Find Probabilities**
* Use a standard normal distribution table or a calculator to find the probabilities associated with the calculated z-scores.
**Important Note:**
* Since the sample size (n) is not given in the problem, we cannot calculate the exact standard error and z-scores.
* Therefore, we cannot provide specific numerical probabilities for parts (a) to (c).
**However, we can explain the general approach:**
**a) Between RM900 and RM1100**
* Calculate the z-scores for RM900 and RM1100 using the formula above.
* Find the area under the standard normal curve between these two z-scores. This will give you the probability that the sample mean falls between RM900 and RM1100.
**b) Below RM900**
* Calculate the z-score for RM900.
* Find the area under the standard normal curve to the left of this z-score. This will give you the probability that the sample mean is below RM900.
**c) Greater than RM850**
* Calculate the z-score for RM850.
* Find the area under the standard normal curve to the right of this z-score. This will give you the probability that the sample mean is greater than RM850.
**d) Exactly RM1000**
* The probability of the sample mean being exactly equal to the population mean (RM1000) is **zero**. This is because the normal distribution is continuous, and the probability of any single point on a continuous distribution is zero.
**In summary:**
* The Central Limit Theorem allows us to approximate the distribution of sample means as normal, even when the population is skewed.
* We need to know the sample size (n) to calculate the standard error and determine the specific probabilities.
* The probability of the sample mean being exactly equal to the population mean is always zero for a continuous distribution.
I hope this explanation helps!
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