Simple random sample of size n=100 obtained from population which has mean of 50 and standard deviation of 40. describe distribution sample of x bar find P(x bar > 105) find P(x bar less than or equal to 98)
Simple random
describe distribution sample of x bar
find P(x bar > 105)
find P(x bar less than or equal to 98)
The Z-score of a random variable X is defined as follows:
Z = (X – µ)/σ.
Here, µ and σ are the mean and standard deviation of X, respectively.
According to central limit theorem, for large sample (>30) the sampling distribution of the sample mean follows approximately normal distribution with the expectation and standard deviation defined as follows:
E(x̅) = µ and SD(x̅) = σ/√n.
Here, n is the sample size, µ is the population mean and σ is the population standard deviation of the random variable X.
Consider a random variable X, which is the interest of the study.
According to the given information, E(X) = 50 and SD (X) = 40.
Now, for sample of 100, the mean of the sampling distribution of the sample mean is,
µx̅ = 50.
Now, for sample of size 100, the standard deviation of the sampling distribution of the sample mean is,
σ x̅ = 40/√100
= 4.
Thus, x̅ has approximately normal with mean 50 and standard deviation 4.
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