Statistics
Statistics
4th Edition
ISBN: 9780393929720
Author: David Freedman, Robert Pisani, Roger Purves
Publisher: Norton, W. W. & Company, Inc.
Question
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Chapter 28.2, Problem 9EA

(a)

To determine

Find the expected balls to be drawn.

(a)

Expert Solution
Check Mark

Answer to Problem 9EA

The expected ball to be drawn is 12.

Explanation of Solution

Calculation:

A set of 10 Ping-Pong balls are considered and there are 120 draws from the bowl.

The expected ball to be drawn is,

Expected=12010=12

Hence, the expected ball to be drawn is 12.

(b)

To determine

Explain how the decisions follow from the data.

(b)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

Chi-square test statistic:

The test statistic for chi-square distribution is,

χ2=Sumof[(observed frequencyexpected frequency)2expected frequency]

With degrees of freedom=number of terms inχ2one

For Ball set A:

The expected values are,

Ball no.

Observed

Frequency

Expected frequency
01312010=12
11112010=12
21612010=12
31112010=12
4512010=12
51212010=12
61212010=12
71912010=12
8512010=12
91612010=12
Total120 

The test statistic is,

χ2=(1312)212+(1112)212+(1612)212++(1612)212=0.08+0.08+1.33++1.33=15.17

Hence, the value of χ2 for Ball set A is 15.17.

For Ball set B:

The expected values are,

Ball no.

Observed

Frequency

Expected frequency
02212010=12
1812010=12
2712010=12
3812010=12
41912010=12
52012010=12
61012010=12
71112010=12
8612010=12
9912010=12
Total120 

The test statistic is,

χ2=(2212)212+(812)212+(712)212++(912)212=8.33+1.33+2.08++0.75=26.67

Hence, the value of χ2 for Ball set B is 26.67.

For Ball set C:

The expected values are,

Ball no.

Observed

Frequency

Expected frequency
01212010=12
11012010=12
21412010=12
31012010=12
41112010=12
51012010=12
62012010=12
71212010=12
81212010=12
9912010=12
Total120 

The test statistic is,

χ2=(1212)212+(1012)212+(1412)212++(912)212=0.00+0.33+0.33++0.75=7.50

Hence, the value of χ2 for Ball set C is 7.50.

For Ball set D:

The expected values are,

Ball no.

Observed

Frequency

Expected frequency
01612010=12
1712010=12
21212010=12
31412010=12
41512010=12
5512010=12
61012010=12
72112010=12
81112010=12
9912010=12
Total120 

The test statistic is,

χ2=(1612)212+(712)212+(1212)212++(912)212=1.33+2.08+0.00++0.75=16.50

Hence, the value of χ2 for Ball set D is 16.50.

There are 10 value. The degrees of freedom is,

degrees of freedom=number of terms inχ2one=101=9

Hence, the degrees of freedom is 9.

From A CHI-SQUARE TABLE, the value corresponding to the 9 degrees of freedom for 10% level is 14.68, 5% level is 16.92, and the 1% level is 21.67.

It is clear that the ball set A and set D are in between the level 1% to 10% indicates that the sets are marginal and it would be retested.

The ball set B test value is greater than the critical values this indicates that the set B is rejected outright, and ball set C test value is less than the critical value which implies that the set C is accepted.

(c)

To determine

Test the Ball set A and D by using retest values.

Comment on the results.

(c)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

For Ball set C:

The expected values are,

Ball no.

Observed

Frequency

Expected frequency
01912010=12
1912010=12
21012010=12
31212010=12
4712010=12
51512010=12
61912010=12
71012010=12
81212010=12
9712010=12
Total120 

The test statistic is,

χ2=(1912)212+(912)212+(1012)212++(712)212=4.08+0.75+0.33++2.08=14.50

Hence, the value of χ2 for Ball set C is 14.50.

For Ball set D:

The expected values are,

Ball no.

Observed

Frequency

Expected frequency
0812010=12
11512010=12
22212010=12
31112010=12
41512010=12
5812010=12
61712010=12
7912010=12
8812010=12
9712010=12
Total120 

The test statistic is,

χ2=(812)212+(1512)212+(2212)212++(712)212=1.33+0.75+8.33++2.08=18.83

Hence, the value of χ2 for Ball set D is 18.83.

The ball set D test value is greater than the critical values of 5% level this indicates that the set D is rejected, and ball set A test value is less than the critical value which implies that the set A is accepted.

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