Answer:
0.0668, 0.0440, 92, 5000, 80
Step-by-step explanation:
a. First calculate the z-score:
z = (x - μ) / σ
z = (71 - 80) / 6
z = -1.5
Now we use a calculator or z-score table to find P(z<-1.5).
P(z<-1.5) = 0.0668
b. Again, we calculate the z-scores:
z = (x - μ) / σ
z = (89 - 80) / 6
z = 1.5
z = (x - μ) / σ
z = (92 - 80) / 6
z = 2
Using a calculator or z-score table, we want to find P(1.5<z<2). Which is equal to P(z<2) - P(z<1.5).
P(z<2) - P(z<1.5)
0.9772 - 0.9332
0.0440
c. This time we want to go backwards. We need to find the z score such that P(z>?) = 0.0250. Or P(z<?) = 1 - 0.0250 = 0.9750. According to the table, z = 1.96.
Now finding the x value that gives us that z-score:
z = (x - μ) / σ
1.96 = (x - 80) / 6
11.76 = x - 80
x = 91.76
Since scores have to be whole numbers, we round up to 92.
d. From part b, we know the z-score for this is 1.5, and that P(z<1.5) = 0.9332. That means that P(z>1.5) = 1 - 0.9332 = 0.0668.
So if 334 scores make up 6.68% of all scores, then the number of students who took the exam is:
0.0668 N = 334
N = 5000
e. For a normal distribution, the median is also the mean. 80.
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