Salaries of travel managers. According to a 2012 Business...

Salaries of travel managers. According to a 2012 Business Travel News survey, the average salary of a travel manager is $\$ 110,550$. Assume that the standard deviation of such salaries is $\$ 25,000$. Consider a random sample of 40 travel managers, and let $\bar{x}$ represent the mean salary for the sample. a. What is $\mu_X$ ? b. What is $\sigma_{\bar{X}}$ ? c. Describe the shape of the sampling distribution of $\bar{x}$. d. Find the $z$-score for the value $\bar{x}=\$ 100,000$. e. Find $P(\bar{x}>100,000)$.

Salaries of travel managers. According to a 2012 Business Travel News survey, the average salary of a travel manager is $\$ 110,550$. Assume that the standard deviation of such salaries is $\$ 25,000$. Consider a random sample of 40 travel managers, and let $\bar{x}$ represent the mean salary for the sample.
a. What is $\mu_X$ ?
b. What is $\sigma_{\bar{X}}$ ?
c. Describe the shape of the sampling distribution of $\bar{x}$.
d. Find the $z$-score for the value $\bar{x}=\$ 100,000$.
e. Find $P(\bar{x}>100,000)$.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is used extensively in statistics, particularly due to the Central Limit Theorem, for approximating the distribution of sample means and for calculating probabilities and critical values in hypothesis testing.

Z-Score

A z-score standardizes a value by expressing the deviation from the mean in units of standard deviation. In the context of sampling distributions, it is used to measure how many standard errors a sample mean is away from the population mean, facilitating probability calculations using the standard normal distribution.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It quantifies the variability of the sample mean around the population mean and is calculated as the population standard deviation divided by the square root of the sample size.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population’s distribution, the distribution of the sample mean will approach a normal distribution as the sample size increases, provided the sample size is sufficiently large. This theorem justifies using normal probability methods for inference on sample means.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of sample means over all possible samples of a fixed size drawn from the population. It provides insights into how the sample mean estimates the population mean and its variability around it.

Transcript

00:01 We're looking at the salaries of travel managers and we're told that the mean for that year of just individual people is $110 ,550 with a sample standard, or excuse me, population standard deviation for individuals.

00:16 So make sure that was neater of $25 ,000.

00:24 And we are taking a sample of 40 of these people and we want to know in part a, what would the mean mean? be of the sampling distribution and the mean would end up being $110 ,550.

00:41 What would the standard deviation be of a sampling mean? well, groups of 40 people finding a mean are not going to have a variability of 25 ,000.

00:51 It will be 25 ,000 divided by the square root of 40 and 25 ,000 divided by the square root of 40.

01:01 Let me get that value.

01:03 That comes out to be $3 ,952 .82, basically.

01:09 And i'm going to store that value in my calculator, so i don't have to type that in again.

01:14 Now, what will be the shape of that distribution? that distribution by the central limit theorem, we have a relatively large sample, large n, therefore this distribution will be approximately normal, or it will be bell -shaped, and it will center at this value, and it will have this standard deviation.

01:37 Now we want to know what would be the z score or the standardized score for someone who makes $100 ,000.

01:46 And we can see it's going to be negative because it's below the mean, and the z will end up being $100 ,000 minus the mean divided by this standard deviation, which i'm just going to mark as that, which is that going to be that amount of money.

02:04 And so let's get that.

02:08 And i guess that $110 ,550 and then divided by that value that i just stored...

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