BenoguigoliB

2021-02-06

1. Going forward, let’s define as the average number of dots after forty rolls.
What is the expected value of x?
The standard deviation of x is 2.197. What is the standard error of x?
2. What is the probability of obtaining an average that is less than 4.25?
$n=40$
$\sigma =2.197$

SoosteethicU

For ques 2, mean is not given so we could not find the probability. I have provided the formula. You need to put the mean and find the probability of P(Z<z) using Normal Distrbution table.
Expected value of average
$E\left(\stackrel{―}{X}\right)=E\left(\sum \frac{{X}_{i}}{n}\right)=\frac{1}{n}×n×E\left(X\right)=E\left(X\right)$
$V\left(\stackrel{―}{X}\right)=V\left(\sum \frac{{X}_{i}}{n}\right)=\frac{1}{{n}^{2}}×n×V\left(X\right)=V\frac{X}{n}$
Standart error of x
${\sigma }_{x}=\sqrt{V}\frac{X}{n}=\frac{{\sigma }_{x}}{\sqrt{n}}$
${\sigma }_{x}=\frac{2.197}{\sqrt{40}}=0.347$
Probability of obtaining an average that is less than 4.25
Z-score is
$Z=\stackrel{―}{x}-\frac{\mu }{\sigma }/\sqrt{n}=4.25-\frac{\mu }{2.197}/\sqrt{40}$

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