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Chapter 10: Q.1.3 (page 608)

Find the probability that p1^-p2^is less than or equal to 0.02. Show your work.

Short Answer

Expert verified

The required probability is 0.2061.

Step by step solution

01

Given Information

Meanμp^1-p^2=-0.10.

Standard deviation σp^1-p^2=0.097.

02

Explanation

The probability that p^1-p^2is less than or equal to -0.02can be computed as:

localid="1650444373982" Pp^1-p^2≥-0.02=Pp^1-p^2-p1-p2σp^1-p^2≥-0.02-p1-p2σp^1-p^2=PZ≥−0.02−(−0.1)0.097=P(Z>0.82)=0.2061

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Min Jae carried out the significance test shown below to answer this question. Unfortunately, he made some mistakes along the way. Identify as many mistakes as you can, and tell how to correct each one.

State: I want to perform a test of

H0:p1-p2=0

Ha:p1-p2>0

where p1=the proportion of Instructor A's students that passed the state exam and p2=the proportion of Instructor B's students that passed the state exam. Since no significance level was stated, I'll use σ=0.05

Plan: If conditions are met, I’ll do a two-sample ztest for comparing two proportions.

Random The data came from two random samples of 50students.

- Normal The counts of successes and failures in the two groups -30,20,22, and 28-are all at least 10.

- Independent There are at least 1000 students who take this driving school's class.

Do: From the data, p^1=2050=0.40and p^2=3050=0.60. So the pooled proportion of successes is

p^C=22+3050+50=0.52

- Test statistic

localid="1650450621864" z=(0.40-0.60)-00.52(0.48)100+0.52(0.48)100=-2.83

- p-value From Table A, localid="1650450641188" P(z≥-2.83)=1-0.0023=0.9977.

Conclude: The p-value, 0.9977, is greater than α=0.05, so we fail to reject the null hypothesis. There is no convincing evidence that Instructor A's pass rate is higher than Instructor B's.
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