$$ H_{0}: p_{1}=p_{2} v s H_{a}: p_{1}p_{2} \text {, } $$ with $$ \hat{p}_{1}-\hat{p}_{2}=0.8-0.7=0.10 $$ for different sample sizes. In parts (a) and (b), use Stat Key or other technology to find the $$ p $$ - value. Click here to access StatKey. (a) $$ \hat{p}_{1}=\frac{24}{30}=0.8 $$ and $$ \hat{p}_{2}=\frac{14}{20}=0.7 $$ Round your answer to two decimal places. p-value $$ = $$ i Are the results significant at the $$ 5 \% $$ level? No $$ v $$ (b) $$ \hat{p}_{1}=\frac{240}{300}=0.8 $$ and $$ \hat{p}_{2}=\frac{140}{200}=0.7 $$ Round your answer to three decimal places. p-value $$ = $$ Are the results significant at the $$ 5 \% $$ level? n

Question

$$ H_{0}: p_{1}=p_{2} v s H_{a}: p_{1}>p_{2} \text {, } $$ with $$ \hat{p}_{1}-\hat{p}_{2}=0.8-0.7=0.10 $$ for different sample sizes. In parts (a) and (b), use Stat Key or other technology to find the $$ p $$ - value. Click here to access StatKey. (a) $$ \hat{p}_{1}=\frac{24}{30}=0.8 $$ and $$ \hat{p}_{2}=\frac{14}{20}=0.7 $$ Round your answer to two decimal places. p-value $$ = $$ i Are the results significant at the $$ 5 \% $$ level? No $$ v $$ (b) $$ \hat{p}_{1}=\frac{240}{300}=0.8 $$ and $$ \hat{p}_{2}=\frac{140}{200}=0.7 $$ Round your answer to three decimal places. p-value $$ = $$ Are the results significant at the $$ 5 \% $$ level? n

UpStudy AI · Accepted Answer

We are testing

H₀: p₁ = p₂  vs.  Hₐ: p₁ > p₂

with the observed difference

p̂₁ – p̂₂ = 0.8 – 0.7 = 0.10.

In both parts the pooled proportion is computed as

p̂ = (x₁ + x₂)⁄(n₁ + n₂).

Below are the steps for each part:

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Part (a):

Data:
  p̂₁ = 24/30 = 0.8  and  p̂₂ = 14/20 = 0.7

Step 1. Compute the pooled proportion:
  p̂ = (24 + 14)⁄(30 + 20) = 38/50 = 0.76

Step 2. Compute the standard error (SE) under H₀:
  SE = √[p̂(1 – p̂) (1/n₁ + 1/n₂)]
    = √[0.76 × 0.24 × (1/30 + 1/20)]
    = √[0.1824 × (0.03333 + 0.05)]
    = √(0.1824 × 0.08333)
    ≈ √(0.01520)
    ≈ 0.1233

Step 3. Compute the test statistic:
  z = (observed difference – 0)⁄SE = 0.10/0.1233 ≈ 0.81

Step 4. Find the p–value for the one–tailed test:
For z ≈ 0.81, the area to the right is
  p–value ≈ 1 − Φ(0.81) ≈ 1 − 0.7910 = 0.2090.
Rounded to two decimal places:
  p–value ≈ 0.21

Step 5. Conclusion:
Since 0.21 > 0.05, the result is not significant at the 5% level.

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Part (b):

Data:
  p̂₁ = 240/300 = 0.8  and  p̂₂ = 140/200 = 0.7

Step 1. Compute the pooled proportion:
  p̂ = (240 + 140)⁄(300 + 200) = 380/500 = 0.76

Step 2. Compute the standard error (SE) under H₀:
  SE = √[0.76 × 0.24 × (1/300 + 1/200)]
    = √[0.1824 × (0.00333 + 0.005)]
    = √[0.1824 × 0.00833]
    ≈ √(0.00152)
    ≈ 0.0390

Step 3. Compute the test statistic:
  z = 0.10/0.0390 ≈ 2.564

Step 4. Find the p–value for the one–tailed test:
For z ≈ 2.56, the area to the right is
  p–value ≈ 1 − Φ(2.56) ≈ 1 − 0.9948 = 0.0052.
Rounded to three decimal places:
  p–value ≈ 0.005

Step 5. Conclusion:
Since 0.005 < 0.05, the result is significant at the 5% level.

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Summary of Answers:

(a) p–value ≈ 0.21. The result is not significant at the 5% level.
(b) p–value ≈ 0.005. The result is significant at the 5% level.
(a) p-value = 0.21. The result is not significant at the 5% level. (b) p-value = 0.005. The result is significant at the 5% level.

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