BASIC PRACTICE OF STATISTICS >C<
BASIC PRACTICE OF STATISTICS >C<
8th Edition
ISBN: 9781319220280
Author: Moore
Publisher: MAC HIGHER
Question
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Chapter 3, Problem 3.28E

(a)

To determine

To obtain: The proportion of observations from a standard normal distribution that falls in z1.63.

To sketch: The standard normal curve for z1.63.

(a)

Expert Solution
Check Mark

Answer to Problem 3.28E

The proportion of observations from a standard normal distribution that falls in z1.63 is 0.0516.

The standard normal curve for z1.63 is

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  1

Explanation of Solution

Calculation:

The z-score less than or equal to –1.63 represents the proportion of observations to the left of −1.63.

Use Table A: Standard normal cumulative proportions to find the area to the left of –1.63.

Procedure:

  • Locate –1.6 in the left column of the table.
  • Obtain the value in the corresponding row below 0.03.

That is, P(z<1.63)=0.0516

Thus, the proportion of observations from a standard normal distribution that falls in z1.63 is 0.0516.

Shade the region to the left of z=1.63 as shown in Figure (1).

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  2

Figure (1)

The shaded region represents the proportion of observations less than or equal to –1.63.

(b)

To determine

To obtain: The proportion of observations from a standard normal distribution that falls in z1.63.

To sketch: The standard normal curve for z1.63.

(b)

Expert Solution
Check Mark

Answer to Problem 3.28E

The proportion of observations from a standard normal distribution that falls in z1.63 is 0.9484.

The standard normal curve for z1.63 is

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  3

Explanation of Solution

Calculation:

The z-score greater than or equal to –1.63 represents the proportion of observations to the right of −1.63. But, Table A: Standard normal cumulative proportions apply only for the cumulative areas to the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of –1.63.

Procedure:

  • Locate –1.6 in the left column of the table.
  • Obtain the value in the corresponding row below 0.03.

That is, P(z<1.63)=0.0516

The area to the right of –1.63 is,

P(z>1.63)=1P(z<1.63)=10.0516=0.9484

Thus, the proportion of observations from a standard normal distribution that falls in z1.63 is 0.9484.

Shade the region to the left of z=1.63 as shown in Figure (2).

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  4

Figure (2)

The shaded region in Figure (2) represents the proportion of observations greater than or equal to –1.63.

(c)

To determine

To obtain: The proportion of observations from a standard normal distribution that falls in z>0.92.

To sketch: The standard normal curve for z>0.92.

(c)

Expert Solution
Check Mark

Answer to Problem 3.28E

The proportion of observations from a standard normal distribution that falls in z>0.92 is 0.1788.

The standard normal curve for z>0.92 is

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  5

Explanation of Solution

Calculation:

The z-score greater than 0.92 represents the proportion of observations to the right of 0.92. But, Table A: Standard normal cumulative proportions apply only for the cumulative areas to the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of 0.92.

Procedure:

  • Locate 0.9 in the left column of the table.
  • Obtain the value in the corresponding row below 0.02.

That is, P(z<0.92)=0.8212

The area to the right of 0.92 is,

P(z>0.92)=1P(z<0.92)=10.8212=0.1788

Thus, the proportion of observations from a standard normal distribution that falls in z>0.92 is 0.1788.

Shade the region to the left of z>0.92 as shown in Figure (3).

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  6

Figure (3)

The shaded region represents the proportion of observations greater than or equal to 0.92.

d)

To determine

To obtain: The proportion of observations from a standard normal distribution that falls in 1.63<z<0.92.

To sketch: The standard normal curve for 1.63<z<0.92.

d)

Expert Solution
Check Mark

Answer to Problem 3.28E

The proportion of observations from a standard normal distribution that falls in 1.63<z<0.92 is 0.7696.

The standard normal curve for 1.63<z<0.92 is

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  7

Explanation of Solution

Calculation:

The z-score between –1.63 and 0.92 represents the proportion of observations to the right of –1.63 and to the left of 0.92.

Use Table A: Standard normal cumulative proportions to find the areas.

Procedure:

For z at –1.63,

  • Locate –1.6 in the left column of the table.
  • Obtain the value in the corresponding row below 0.03.

That is, P(z<1.63)=0.0516

For z at 0.92,

  • Locate 0.9 in the left column of the table.
  • Obtain the value in the corresponding row below 0.02.

That is, P(z<0.92)=0.8212

Hence, the difference between the areas to the left of –0.42 and the left of 2.12 is,

P(1.63<z<0.92)=P(z<0.92)P(z<1.63)=0.82120.0516=0.7696

Thus, the proportion of observations from a standard Normal distribution that takes values between –1.63 and 0.92 is 0.7696.

Shade the region to the right of z=1.63 to the left of z=0.92 as shown in Figure (4).

BASIC PRACTICE OF STATISTICS >C<, Chapter 3, Problem 3.28E , additional homework tip  8

Figure (4)

The shaded region represents the proportion of observations between –1.63 and 0.92

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