Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, is less than 78 is:
a. 0.9332
b. 0.5987
c. 1.5000
d. 0.0668

a. 0.9332

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Let X the random variable of interest. We know from the problem that the distribution for the random variable X is given by:

We take a sample of n=36 . That represent the sample size.

What can we say about the shape of the distribution of the sample mean time?

From the central limit theorem we know that the distribution for the sample mean is also normal and is given by:

What is the standard error of the mean time?

The standard error is given by this formula:

What percent of the sample means will be greater than 78?

In order to answer this question we can use the z score in order to find the probabilities, the formula given by:

And we want to find this probability:

a. 0.9332

wheres graph

step-by-step explanation:

step-by-step explanation:

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