Teachers’ Salaries in North Dakota: The average teacher’s salary in North Dakota is 35,441. Assume a normal distribution with $\sigma =5100$.

What is the probability that a randomly selected teacher’s salary is greater than $43,700?

For a sample of 90 teachers, what is the probability that the sample mean is greater than 36,917? Assume that the sample is taken from a large population and the correction factor can be ignored.

Answer :

Given :

The population mean $(\mu )=35,441$

The population standard deviation $(\sigma )=5,100$

Solution :

a) The Probability of a Teacher’s Salary Being Greater Than $43,700 :

$$\therefore P(x>43,700)=P(\frac{x–\mu }{\sigma }>\frac{43,700–35,441}{5,100})$$ $$=P(z>1.62)$$ $$=0.0526$$

b) The Probability of Sample Mean Salary Being Greater Than $36,917 :

→ here we have The sample size (n)=90

$$\therefore P(\overline{x}>36,917)=P(\frac{\overline{x}–\mu }{\sigma /\sqrt{n}}>\frac{36,917–35,441}{\frac{5,100}{\sqrt{90}}})$$ $$=P(z>2.75)$$ $$=0.0030$$