Salaries for various positions can vary significantly,...

Salaries for various positions can vary significantly, depending on whether or not the company is in the public or private sector. The U.S. Department of Labor posted the 2007 average salary for human resource managers employed by the federal government as $ 76,503 .$ Assume that annual salaries for this type of job are normally distributed and have a standard deviation of $8850$ a. What is the probability that a randomly selected human resource manager received over $100,000$ in $2007 ?$ b. $\quad$ A sample of 20 human resource managers is taken and annual salaries are reported. What is the probability that the sample mean annual salary falls between $70,000$ and $80,000 ?$

Salaries for various positions can vary significantly, depending on whether or not the company is in the public or private sector. The U.S. Department of Labor posted the 2007 average salary for human resource managers employed by the federal government as $ 76,503 .$ Assume that annual salaries for this type of job are normally distributed and have a standard deviation of $8850$
a. What is the probability that a randomly selected human resource manager received over $100,000$ in $2007 ?$
b. $\quad$ A sample of 20 human resource managers is taken and annual salaries are reported. What is the probability that the sample mean annual salary falls between $70,000$ and $80,000 ?$

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is fully described by its mean and standard deviation, making it a fundamental model for natural events and measurement errors.

Z-Score

A Z-score is a standardized value that indicates how many standard deviations an element is from the mean. This concept allows us to transform data from a normal distribution into a standard normal distribution, facilitating the calculation of probabilities and comparisons between different normal distributions.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means of a given size drawn from the population. When the population is normally distributed, the sample mean is also normally distributed regardless of sample size, which makes it easier to make probabilistic inferences about the population mean.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean. It quantifies the variability of the sample mean around the population mean and decreases as the sample size increases, providing an indication of the accuracy of the sample mean as an estimate of the population mean.

Transcript

00:02 Okay, for this problem, we're given some human resource data about salaries and jobs.

00:07 So annual salaries for this job are supposed to be, on average, $76 ,503.

00:17 And the variation, or more precisely, the standard deviation from this 2007 data says those salaries go up and down, or the typical deviation from the mean is $88 ,8 ,850.

00:33 Dollars.

00:34 Okay.

00:35 And as with any problem, we should probably kind of sketch out what it looks like, because it's normally distributed.

00:41 And what we're doing with this problem is really we're just looking at some likelihoods of what occurs of a single value and the probability of something occurring for more of a sample mean.

00:57 So let's look at this for the single values.

00:59 So what's the probability? we're going to translate this and write it into our probability notation.

01:03 So we're the probability that the branch that the human resources person earns over 100 ,000.

01:13 All right.

01:14 So the reason i like to draw it, it kind of gives you a general idea.

01:17 So really, if this is about $76 ,000 for the sample mean, $100 ,000 is going to be way over here.

01:26 So it's going to be way after the side.

01:27 So the probability is pretty likely that a single person does, but it's possible.

01:32 So let's see what it is.

01:33 I'm going to use our calculator.

01:34 We've done enough of these now that we're going to take a look.

01:37 And let the calculator look at our distributions.