Two normal distributions in the form [math]\mbox{N}(\mu,\sigma^2)[/math] are shown.[br][br]The blue curve is [math]X \sim \mbox{N}(1,0)[/math] and the black curve is [math] \mbox{N}(\mu,\sigma^2)[/math] with parameters controlled by the input boxes.[br][br]The shaded areas represent the probabilities shown, you can move the points to change the probabilities.

[b]Tasks[/b][br][list=1][br][*] Set the black curve to [math]X \sim \mbox{N}(4,1)[/math] and find [math]P(X\le 5)[/math].[br][br][*] Find the value of [math]z[/math] such that [math]P(X\le 5)=P(Z\le z) [/math].[br][br][*] Set the black curve to [math]X \sim \mbox{N}(6,1)[/math] and find [math]P(X\le 4)[/math].[br][br][*] Find the value of [math]z[/math] such that [math]P(X\le 4)=P(Z\le z) [/math].[br][br][*] What do you notice?[br][br][*] Set the black curve to [math]X \sim \mbox{N}(0,4)[/math] and find [math]P(X\le 4)[/math].[br][br][*] Find the value of [math]z[/math] such that [math]P(X\le 4)=P(Z\le z) [/math].[br][br][*] Set the black curve to [math]X \sim \mbox{N}(0,9)[/math] and find [math]P(X\le 6)[/math].[br][br][*] Find the value of [math]z[/math] such that [math]P(X\le 6)=P(Z\le z) [/math].[br][br][*] What do you notice?[br][br][*] Test your findings by trying to find [math]P(X\le 5) [/math] if [math]X \sim \mbox{N}(4,6)[/math] using the red area only, then check your answer.[br][/list][br]Created by Dr GJ Daniels