Answer:
1)
a) 0.9706 = 97.06% probability that x is less than 60.
b) 0.9987 = 99.87% probability that x is greater than 16.
c) 0.9693 = 96.93% probability that x is between 16 and 60.
d) 0.0294 = 2.94% probability that x is more than 60.
2)
a) 0.0668 = 6.68% probability that the thickness is less than 3.0 mm.
b) 0.0062 = 0.62% probability that the thickness is more than 7.0 mm
c) 0.9270 = 92.70% probability that the thickness is between 3.0 mm and 7.0 mm.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
1)
We have that ![\mu = 43, \sigma = 9](https://tex.z-dn.net/?f=%5Cmu%20%3D%2043%2C%20%5Csigma%20%3D%209)
a) x is less than 60
This is the pvalue of Z when X = 60.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{60 - 43}{9}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B60%20-%2043%7D%7B9%7D)
![Z = 1.89](https://tex.z-dn.net/?f=Z%20%3D%201.89)
has a pvalue of 0.9706
0.9706 = 97.06% probability that x is less than 60.
b) x is greater than 16
This is 1 subtracted by the pvalue of Z when X = 16.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{16 - 43}{9}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B16%20-%2043%7D%7B9%7D)
![Z = -3](https://tex.z-dn.net/?f=Z%20%3D%20-3)
has a pvalue of 0.0013.
1 - 0.0013 = 0.9987
0.9987 = 99.87% probability that x is greater than 16.
c) x is between 16 and 60
This is the pvalue of Z when X = 60 subtracted by the pvalue of Z when X = 16.
From a), Z when X = 60 has a pvalue of 0.9706.
From b), Z when X = 16 has a pvalue of 0.0013
0.9706 - 0.0013 = 0.9693
0.9693 = 96.93% probability that x is between 16 and 60.
d) x is more than 60
This is 1 subtracted by the pvalue of Z when X = 60.
From a), Z when X = 60 has a pvalue of 0.9706.
1 - 0.9706 = 0.0294
0.0294 = 2.94% probability that x is more than 60.
2)
Now ![\mu = 4.5, \sigma = 1](https://tex.z-dn.net/?f=%5Cmu%20%3D%204.5%2C%20%5Csigma%20%3D%201)
a) the thickness is less than 3.0 mm
This is the pvalue of Z when X = 3.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{3 - 4.5}{1}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B3%20-%204.5%7D%7B1%7D)
![Z = -1.5](https://tex.z-dn.net/?f=Z%20%3D%20-1.5)
has a pvalue of 0.0668
0.0668 = 6.68% probability that the thickness is less than 3.0 mm.
b) the thickness is more than 7.0 mm
This is 1 subtracted by the pvalue of Z when X = 7.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{7 - 4.5}{1}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B7%20-%204.5%7D%7B1%7D)
![Z = 2.5](https://tex.z-dn.net/?f=Z%20%3D%202.5)
has a pvalue of 0.9938.
1 - 0.9938 = 0.0062
0.0062 = 0.62% probability that the thickness is more than 7.0 mm
c) the thickness is between 3.0 mm and 7.0 mm.
This is the pvalue of Z when X = 7 subtracted by the pvalue of Z when X = 3.
From b), Z when X = 7 has a pvalue of 0.9938.
From a), Z when X = 3 has a pvalue of 0.0668
0.9938 - 0.0668 = 0.9270
0.9270 = 92.70% probability that the thickness is between 3.0 mm and 7.0 mm.