Question

In this exercise we examine the effect of the sample size on the significance test for comparing two proportions. In each case suppose that p̂1 = 0.65 and p̂2 = 0.45, and take n to be the common value of n1 and n2. Use the z statistic to test H0: p1 = p2 versus the alternative Ha: p1 ≠ p2. Compute the statistic and the associated P-value for the following values of n: 40, 50, 60, 80, 380, 480, and 980. Summarize the results in a table. (Test the difference p1 − p2. Round your values for z to two decimal places and round your P-values to four decimal places.)

n z P-value

40

50

60

80

380

480

980

Explain what you observe about the effect of the sample size on statistical significance when the sample proportions p̂1 and p̂2 are unchanged.

As sample size increases, the test becomes less significant.

As sample size increases, the test becomes more significant.

As sample size increases, there is no effect on significance.

There is not enough information.

Answer 1

SOLUTION:

From given data,

In this exercise we examine the effect of the sample size on the significance test for comparing two proportions. In each case suppose that p̂₁ = 0.65 and p̂₂ = 0.45, and take n to be the common value of n1 and n2. Use the z statistic to test H0: p₁ = p₂ versus the alternative Ha: p₁ ≠ p₂. Compute the statistic and the associated P-value for the following values of n: 40, 50, 60, 80, 380, 480, and 980.

p̂₁ = 0.65

p̂₂ = 0.45

The value of z for the sample size as 40 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 40 ) + (0.45 (1-0.45) / 40 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65) / 40 ) + (0.45 (1-0.45) / 40 ) )

z = 0.2 / 0.10897247

z = 1.83

The p-value for two tailed ,

P(z=1.83) = 2*P(z > 1.83)

= 2* (1-P(z < 1.83))

= 2* (1- 0.96638)

= 2*0.03362

= 0.06724

The value of z for the sample size as 50 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 50 ) + (0.45 (1-0.45) / 50 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65​​​​​​​) / 50 ) + (0.45 (1-0.45) / 50 ) )

z = 0.2 / 0.09746794

z = 2.05

The p-value for two tailed ,

P(z=2.05) = 2*P(z > 2.05)

= 2* (1-P(z < 2.05))

= 2* (1- 0.97982)

= 2*0.02018

= 0.04036

The value of z for the sample size as 60 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 60 ) + (0.45 (1-0.45) / 60 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65​​​​​​​) / 60 ) + (0.45 (1-0.45) / 60 ) )

z = 0.2 / 0.08897565

z =2.24

The p-value for two tailed ,

P(z=2.24) = 2*P(z > 2.24)

= 2* (1-P(z < 2.24))

= 2* (1- 0.98745)

= 2*0.01255

= 0.0251

The value of z for the sample size as 80 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 80 ) + (0.45 (1-0.45) / 80 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65​​​​​​​) / 80 ) + (0.45 (1-0.45) / 80 ) )

z = 0.2 / 0.07705517

z =2.59

The p-value for two tailed ,

P(z=2.59) = 2*P(z > 2.59)

= 2* (1-P(z <2.59))

= 2* (1- 0.99520)

= 2*0.0048

= 0.0096

The value of z for the sample size as 380 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 380 ) + (0.45 (1-0.45) / 380 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65​​​​​​​) / 380 ) + (0.45 (1-0.45) / 380 ) )

z = 0.2 / 0.03535533

z =5.65

The p-value for two tailed ,

P(z=5.65) = 2*P(z > 5.65)

= 2*0

=0

The value of z for the sample size as 480 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 480 ) + (0.45 (1-0.45) / 480 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65​​​​​​​) / 480 ) + (0.45 (1-0.45) / 480 ) )

z = 0.2 / 0.03145764

z =6.35

The p-value for two tailed ,

P(z=6.35) = 2*P(z > 6.35)

= 2*0

=0

The value of z for the sample size as 980 is obtained as shown below:

z = (p̂₁ - p̂₂) / sqrt ( (p̂₁ (1-p̂₁) / n ) + (p̂₂ (1-p̂₂) / n ) )

z = (0.65 - 0.45) / sqrt ( (0.65 (1-0.65) / 980 ) + (0.45 (1-0.45) / 980 ) )

z = 0.2 / sqrt ( (0.65 (1-0.65​​​​​​​) / 980 ) + (0.45 (1-0.45) / 980 ) )

z = 0.2 / 0.02201576

z =9.08

The p-value for two tailed ,

P(z=9.08) = 2*P(z > 9.08)

= 2*0

=0

Please thumbs-up / vote up this answer if it was helpful. In case of any problem, please comment below. I will surely help. Down-votes are permanent and not notified to us, so we can't help in that case.