Chapter 2.2, Problem 47E

(a)

To determine

To find: the proportionof observations from the standard Normal distribution

Answer to Problem 47E

The proportion is read as .9978

Explanation of Solution

Given:

  $z<2.85$

Calculation:

We are to use the Standard Normal table to calculate each proportion corresponding to a givenz value. We use the notation P( ) to designate that we are interested in computing a probability or a proportion.

P(z < 2.85). Here is the sketch of the desired proportion, with the area of interest shaded:

  

From the Standard Normal table, we look up the z-value 2.8 on the left side of the table, and then move to the right in that row until we are under the column headed by 05. The proportion is read as .9978

Conclusion:

Hence,the proportion is read as .9978

(b)

To determine

To find: the proportionof observations from the standard Normal distribution

Answer to Problem 47E

The area to the right of z = 2.85 is .0022

Explanation of Solution

Given:

  $z>-1.66$

Calculation:

P (z > 2.85). This is the "opposite" of the proportion that we found in part (A). We shade theother side of the standard normal curve, as follows:

  

Conclusion:

Hence, the area to the right of z = 2.85 is .0022

(c)

To determine

To find: the proportionof observations from the standard Normal distribution

Answer to Problem 47E

We see that P (z > -1.66) is .9515.

Explanation of Solution

Given:

  $z>-1.66$

Calculation:

P (z > -1.66). Here is the sketch of the desired proportion:

  

From the Standard Normal table, we look up the z-value -1.6 on the left side of the table, and then move to the right in that row until we are under the column headed by .06. The proportion is read as .0485. But this is the proportion to the left of -1.66. To calculate the proportion to theright of -1.66, we simply subtract from 1, as shown here:

area to right = 1- .0485

=.9515

We see that $P(z>-1.66)\text{ is }\text{.9515}$

Conclusion:

Hence, we see that $P(z>-1.66)\text{ is }\text{.9515}$

(d)

To determine

To find: the proportionof observations from the standard Normal distribution

Answer to Problem 47E

Area (proportion) B is .9493, which is the P (-1.66 < z < 2.85).

Explanation of Solution

Given:

  $-1.66<z<2.85$

Calculation:

P (-1.66 < z < 2.85). This is a "between" problem, and it requires multiple steps to solve. Wefirst prepare the sketch:

  

We now look at the sketch again, and we identify three different pieces of it: the unshaded piece to the left of -1.66, the shaded piece between -1.66 and 2.85, and the unshaded piece to the right of 2.85. We designate them as areas A. B, and C, respectively:

  

In part (C) we determined that area A was .0485, and in part (B) we found that area C was0022.

We know that the total area, or proportion, under the curve is 1. Thus, we know that A +B +C = 1. We know A and C already. So it is a matter of subtraction to determine the value of area B, which is the area (proportion) of interest.

A+B+C=1

.0485+B+.0022=1

B+.0507=1

B=.9493

Area (proportion) B is .9493, which is the P (-1.66 < z < 2.85).

Conclusion:

Hence, area (proportion) B is .9493, which is the P (-1.66 < z < 2.85).