Please answer these 2 quesitions for this problem... 1. Identify if the Central
ID: 3064695 • Letter: P
Question
Please answer these 2 quesitions for this problem...
1. Identify if the Central Limit Theorem applies for the case or not.
2. Describe the mean and standard error of the sampling distribution of the means.
13. A population has a mean of µ=30 and a standard deviation of =8.
a) If the population distribution is normal, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=7?
b) If the population distribution is positively skewed, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=7?
c) If the population distribution is normal, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=67?
d) If the population distribution is positively skewed, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=67
Please answer these 2 quesitions for this problem...
1. Identify if the Central Limit Theorem applies for the case or not.
2. Describe the mean and standard error of the sampling distribution of the means.
Explanation / Answer
13. Given that,
Population mean (µ) = 30
and standard deviation () = 8
a) If the population distribution is normal, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=7?
We have to find P(Xbar > 32)
Convert Xbar = 32 into z-score.
z = (Xbar - mu) / (sigma / sqrt(n))
= (32 - 30) / (8 / sqrt(7)) = 0.66
Now we have to find P(Z > 0.66)
This probability we can find in excel.
syntax :
=1 - NORMSDIST(z)
where z is z-score.
P(Z > 0.66) = 0.2542
b) If the population distribution is positively skewed, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=7?
If the underlying distribution is positively skewed, the distribution of sample means (for large n) will tend to be normal distribution.
We get same probability as a).
c) If the population distribution is normal, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=67?
Now here we have to find P(Xbar > 32)
z-score for Xbar = 32 is,
z = (32 - 30) / (8 / sqrt(67)) = 2.05
P(Z > 0.9) = 0.0204
d) If the population distribution is positively skewed, what is the probability of obtaining a sample mean greater than M=32 for a sample of n=67
If the underlying distribution is positively skewed, the distribution of sample means (for large n) will tend to be normal distribution.
We get same probability as d).
We can use central limit theorem when :
The samples must be independent
The sample size must be “big enough"
Mean of the sampling distribution of mean is population mean and
standard error = sigma / sqrt(n)
Central limit theorem states that for large n the distribution of sample mean goes to normal iff
mean is population mean
standard error = sigma / sqrt(n)