The average teacher's salary in Connecticut (ranked first...

The average teacher's salary in Connecticut (ranked first among states) is $\$ 57,337$. Suppose that the distribution of salaries is normal with a standard deviation of $\$ 7500 .$ a. What is the probability that a randomly selected teacher makes less than $\$ 52,000$ per year? b. If we sample 100 teachers' salaries, what is the probability that the sample mean is less than $\$ 56,000 ?$

The average teacher's salary in Connecticut (ranked first among states) is $\$ 57,337$. Suppose that the distribution of salaries is normal with a standard deviation of $\$ 7500 .$
a. What is the probability that a randomly selected teacher makes less than $\$ 52,000$ per year?
b. If we sample 100 teachers' salaries, what is the probability that the sample mean is less than $\$ 56,000 ?$

Key Concepts

Central Limit Theorem (Sampling Distribution of the Mean)

The Central Limit Theorem states that the distribution of the sample mean approximates a normal distribution as the sample size becomes large, regardless of the original distribution of the data, provided the samples are independent and identically distributed. This concept is crucial when making inferences about population parameters based on sample data, particularly in calculating probabilities concerning sample means.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by its mean and standard deviation and is used to represent real-valued random variables whose distributions are not known. Many natural phenomena can be approximated by this distribution, making it fundamental in statistics for modeling and inference.

Standardization (Z-Score)

Standardization is the process of converting a normal random variable to a standard normal variable by subtracting the mean and dividing by the standard deviation. The resulting value, known as the z-score, indicates how many standard deviations the original value is from the mean. This technique is used to compare values from different normal distributions or to find probabilities associated with specific outcomes.

Transcript

00:01 In this problem, we have two parts.

00:03 We have an a and a b part.

00:05 And both parts are centered around the fact that we have an average teacher salary.

00:11 So we're talking about the population of teachers in connecticut, having an average salary of $57 ,337.

00:28 And the standard deviation is $7 ,500.

00:33 So we are going to use that in both parts.

00:42 So let's focus on part a.

00:46 And in part a, we want the probability that a teacher, one teacher, has a salary that's less than $52 ,000 a year.

01:05 So with that, we are going to have to draw a bell curve.

01:17 And with the bell curve, we always put the average in the center.

01:20 So we're going to have 57 ,337 in the center.

01:27 And we are looking for an average, or sorry, a salary, less than 52 ,000.

01:40 So the next thing we're going to have to do is we're going to have to calculate the z score associated with 52 ,000.

01:49 So we're going to do 52 ,000 minus 57 ,337, divide it by.

02:00 The standard deviation of 7 ,500, and the z score associated with the 52 ,000 is approximately negative .71.

02:15 So what we can do is we can go back to our picture, and we could say that 52 ,000 is equivalent to a z score of negative .71.

02:26 So we can also rewrite our problem to say the probability that the z score is the z score is less than negative.

02:34 0 .71.

02:38 So you're then going to use your standard normal distribution chart in the back of your textbook.

02:42 And when we look up negative 0 .71, we are going to get a value of 0 .2389.

02:58 So for part a, the probability that a teacher selected has a salary of less than $52 ,000 would be 0 .2389...

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