A population has a mean of 75 and a standard deviation of 8. A random sample of 800 is selected. The expected value of LaTeX: \bar{x}x ¯ is

a.8

b.75

c.800

d. None of these alternatives is correct.

This question is based on the concept of statistics.Therefore, the expected value of  mean is 75. Hence, the correct option is (b) 75.

Given:

Mean = 75

Standard deviation = 8

Random sample size = 800

According to the question,

By using the central limit theorem states that,

This theorem states that, the distribution of sample means approximate normal distribution as the sample size gets larger.

Hence, for a skewed variable, the central limit theorem can also be applied, as long as n is at least 30.

By the above theorem, the mean of the sample, is expected to be also 75.

Therefore, the expected value of  mean is 75. Hence, the correct option is (b) 75.

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b.75

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem:

Mean of the population is 75.

By the Central Limit Theorem,

The mean of the sample, , is expected to be also 75.

So the correct answer is:

b.75

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step-by-step explanation: