Solved on Nov 02, 2023

Probability that a randomly selected teacher in Connecticut makes less than 54,500peryear.54,500 per year. P(X<54,500)=Φ(54,50057,3377,500) P(X < 54,500) = \Phi \left( \frac{54,500 - 57,337}{7,500} \right) \approx \square $

STEP 1

Assumptions1. The average teacher's salary in Connecticut is \$57,337. The distribution of salaries is normal3. The standard deviation of the salaries is \$7,5004. We want to find the probability that a randomly selected teacher makes less than \$54,500 per year

STEP 2

To find the probability that a randomly selected teacher makes less than \54,500peryear,weneedtoconvertthesalarytoazscore.Thezscoreisameasureofhowmanystandarddeviationsanelementisfromthemean.Theformulaforthezscoreis54,500 per year, we need to convert the salary to a z-score. The z-score is a measure of how many standard deviations an element is from the mean. The formula for the z-score isz=xμσz = \frac{x - \mu}{\sigma}wherewhere- xisthevalueweareinterestedin(inthiscase,$54,500) is the value we are interested in (in this case, \$54,500) - \muisthemean(inthiscase,$57,337) is the mean (in this case, \$57,337) - \sigma$ is the standard deviation (in this case, \$7,500)

STEP 3

Plug in the values for xx, μ\mu, and σ\sigma into the z-score formula.
z=$54,500$57,337$7,500z = \frac{\$54,500 - \$57,337}{\$7,500}

STEP 4

Calculate the z-score.
z=$54,500$57,337$7,500=0.38z = \frac{\$54,500 - \$57,337}{\$7,500} = -0.38

STEP 5

Now that we have the z-score, we can find the probability that a randomly selected teacher makes less than \$54,500 per year. This is the same as finding the area to the left of the z-score on the standard normal distribution. We can use a z-table or a calculator to find this probability.
(X<$54,500)=(Z<0.38)(X<\$54,500) =(Z<-0.38)

STEP 6

Look up the probability associated with the z-score in a z-table or use a calculator to find the probability.
(Z<0.38)=0.3520(Z<-0.38) =0.3520So, the probability that a randomly selected teacher makes less than \$54,500 per year is0.3520 or35.20%.

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