In: Statistics and Probability
Left endpoint: right endpoint:
Q1 =
Solution(a)
We need to calculate Z-scores bound the middle 94% of the standard
normal distribution i.e. P(-z<Z<z) = 0.94
we need to calculate z
Here alpha = 0.94
we need to calculate middle 94% percent so
p-value for upper bound = 0.97
p-value for lower bound = 0.03
From Z table we found Z-score for lower bound = -1.88
From Z table we found Z-score for upper bound = +1.88
So P(-1.88<Z<1.88) = 0.94
Solution(b)
Given in the question
Mean()
= 9.39
Standard deviation()
= 1.7
We need to calculate Q1 or 25th percentile
So p-value = 0.25, From Z table we found Z-score = -0.6745
So Q1 can be calculated as
Q1 =
+ Z-score *
= 9.39 - 0.6745*1.7 = 8.24
Q1 = 8.24
Solution(c)
Given in the question
Mean()
= 85.84
Standard deviation()
= 12.95
We need to calculate Q1 or 32th percentile
So p-value = 0.32, From Z table we found Z-score = -0.4677
So 32 percentile can be calculated as
X =
+ Z-score *
= 85.84 - 0.4677*12.95 = 79.7833
32 percentile is 79.7833
Solution(d)
Given in the question
Mean()
= 71.66
Standard deviation()
= 8.46
We need to calculate the lowest score a student could earn and
still be in the top 10%
So p-value = 0.90, From Z table we found Z-score = 1.28155
So Score can be calculated as
X =
+ Z-score *
= 71.66 + 1.28155*8.46 = 82.502 or 83
lowest score is 83 a student could earn and still be in the top
10%
Solution(e)
Given in the question
Mean()
= 470
Standard deviation()
= 108
We need to calculate the lowest score for 80th percentile
So p-value = 0.80, From Z table we found Z-score = 0.8416
So Score can be calculated as
X =
+ Z-score *
= 470 + 0.8416*108 = 560.89 or 561
80th percentile lowest score = 561
Solution(f)
Mean()
= 21.3
Standard deviation()
= 5.3
We need to calculate 96th percentile for the heights which can be
calculated as
So p-value = 0.96, From Z table we found Z-score = 1.75
So Score can be calculated as
X =
+ Z-score *
= 21.3 + 1.75*5.3 =30.575
96th percentile height = 30.575
Solution(g)
Mean()
= 8.5
Standard deviation()
= 0.28
We need to calculate 25th percentile so that we can found
Three-quarters of the pH measurements in this river basin are
greater than
So p-value = 0.25, From Z table we found Z-score = -0.6745
So Score can be calculated as
X =
+ Z-score *
= 8.5 - 0.6745*0.28 =8.31
Three-quarters of the pH measurements in this river basin are
greater than 8.31