Consider the normally distributed graph with a standard deviation of 1 and a mea
ID: 3043825 • Letter: C
Question
Consider the normally distributed graph with a standard deviation of 1 and a mean of 0:
Set the z-scores to -1.0 and +1.0. What is the area in the middle bounded by these z-scores? What is the area bounded by -2, +2 and -3, +3?
What percentage of a normal distribution lies between a z-score of-2 and +2?
Set the area in the middle to 0.95? What z-scores bound this area?
What proportion of the normal distribution would have values greater than a z-score of 1.96?
Now just try highlighting either the left or right hand side of the curves and look at how the percentages change. What happens when you set the area to 0.5? What does the graph look like?
What is the proportion of the distribution greater than z = 1.65?
What is the proportion of the distribution less than z = 1.65?
What is the proportion of the distribution less than z = -2?
What is the proportion of the distribution greater than z = -1.45?
What proportion of the distribution would lie to the left and right of z-scores of -1.96 and a z-score of +1.96?
.
If the distribution was not normal (imagine bimodal or trimodal), would the proportion of the distribution still be the same as the normal distribution?
.
Normal Probability Applet Area 0.3173 -4 2 z-score -1.00 1.000 Z-scoreExplanation / Answer
R codes provided along with the answer.
Q. What is the area in the middle bounded by these z-scores?
> pnorm(1)-pnorm(-1)
[1] 0.6826895
Q. What is the area bounded by -2, +2 and -3, +3?
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> pnorm(3)-pnorm(-3)
[1] 0.9973002
Q. What percentage of a normal distribution lies between a z-score of-2 and +2?
0.9544997*100 = 95.45%
Q. Set the area in the middle to 0.95? What z-scores bound this area?
> qnorm(0.95)
[1] 1.644854
(0 , 1.6444854)
Q. What proportion of the normal distribution would have values greater than a z-score of 1.96?
> pnorm(1.96,lower.tail=F)
[1] 0.0249979
2.5% of the distribution.
Q. What is the proportion of the distribution greater than z = 1.65?
> pnorm(1.65,lower.tail=F)
[1] 0.04947147
4.95% of the distribution.
Q. What is the proportion of the distribution less than z = 1.65?
> pnorm(1.65,lower.tail=T)
[1] 0.9505285
95.05%.
Q. What is the proportion of the distribution less than z = -2?
> pnorm(-2)
[1] 0.02275013
2.27%.
Q. What is the proportion of the distribution greater than z = -1.45?
> pnorm(-1.45,lower.tail=F)
[1] 0.9264707
92.65%.
Q. What proportion of the distribution would lie to the left and right of z-scores of -1.96 and a z-score of +1.96?
> pnorm(-1.96)+pnorm(1.96,lower.tail=F)
[1] 0.04999579
4.9996%.
If the distribution was non-normal, the proportion would have changed.
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