A random sample of size 64 is taken from a normal population with μ=51.4 and σ=6.8 . What is the

step 1 So we have a sample size of 64 taken from a normal population. So we'll have a new sampling dist

A random sample of size 64 is taken from a normal population...

A random sample of size 64 is taken from a normal population with $\mu=51.4$ and $\sigma=6.8 .$ What is the probability that the mean of the sample will
(a) exceed $52.9$;
(b) fall between $50.5$ and $52.3$;
(c) be less than $50.6 ?$

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Key Concepts

Sampling Distribution of the Sample Mean

This concept describes the probability distribution of sample means when samples are drawn from the same population. For a normally distributed population (or in large samples by the Central Limit Theorem), the sample mean is normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It quantifies the variability inherent in the sample mean due to sampling variability and is computed as the population standard deviation divided by the square root of the sample size.

Z-Score

A z-score is a way to standardize a value from a normal distribution, indicating how many standard deviations that value is from the mean. By converting to a z-score using the formula (value minus mean) divided by the standard error, probabilities can be determined using the standard normal distribution.

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