Answer to Question #207752 in Statistics and Probability for Mary

Question

A. Use the z-table to find the area that corresponds to each of the following:

1. z = 0.67

2. z = − 1.5

3. z = .78

4. z = − 2.35

5. z = 1.8

B. Given = 62 and = 8. Find the z-score value that corresponds to each of the following scores.

1. X = 50

2. X = 78

3. X = 82

Find the raw score that corresponds to each of the following.

4. = 8, = 52, z = − 1.5

5. = 15, = 75, z = 0.47

C. Solving Problem.

1. The length of human pregnancies from conception to birth approximates a normal 

distribution with a mean of 266 days and a standard deviation of 16 days. What 

probability/proportion of all pregnancies will last between 240 and 270 days (roughly 

between 8 and 9 months)?

2. The result of a nationwide aptitude test in mathematics are normally distributed with = 80 

and = 15. 

a. What is the percentile rank of a score of 87?

b. What is the score that corresponds to a percentile rank of 94.5%?

Answer 1

A.

1. $P(z<0.67)=0.7486$

2. $P(z<-1.5)=0.0668$

3. $P(z<0.78)=0.7823$

4. $P(z<-2.35)=0.0094$

5. $P(z<1.8)=0.9641$

B.

1. $z=\dfrac{50-62}{8}=-1.5$

2. $z=\dfrac{78-62}{8}=2$

3. $z=\dfrac{82-62}{8}=2.5$

4. $x=\mu+z\sigma=52-1.5(8)=40$

5. $x=\mu+z\sigma=75+0.47(15)=82.05$

C.

1.

$$P(240<X<270)=P(X<270)-P(X\leq240)$$

$$=P(Z<\dfrac{270-266}{16})-P(Z\leq\dfrac{240-266}{16})$$

$$=P(Z<0.25)-P(Z\leq-1.625)$$

$$\approx0.59871-0.05208\approx0.5466, 54.66\%$$

2.

a.

$$P(X<87)=P(Z<\dfrac{87-80}{15})$$

$$\approx P(Z<0.46667)\approx0.6796$$

$$\approx0.59871-0.05208\approx0.5466, 54.66\%$$

b.

$$P(Z<z)=0.945$$

$$z\approx1.5982\approx1.6$$

$$x\approx80+1.5982(15)\approx103.973\approx104$$