Questions: Suppose that for fast food restaurant A, it was found that 264 orders were accurate and 54 orders were not level to test the claim B, 329 orders were accurate and 33 orders were not accurate. Use a 0.05 significance claim that restaurants A B have the same accuracy rates. Let P1= prop of restaurant A orders (accurate) P2= prop of restaurant B orders (accurate) Claim: P1=P2 (H0) Counterclaim: P1 != P2 (H1) "two tailed test" p̄=(x1+x2)/(n1+n2)=(264+329)/(318+362)=593/680 q̄=1-(593/680)=87/680

Suppose that for fast food restaurant A, it was found that 264 orders were accurate and 54 orders were not level to test the claim B, 329 orders were accurate and 33 orders were not accurate. Use a 0.05 significance claim that restaurants A  B have the same accuracy rates.
Let P1= prop of restaurant A orders (accurate)
P2= prop of restaurant B orders (accurate)
Claim: P1=P2 (H0) Counterclaim: P1 != P2 (H1)
"two tailed test"
p̄=(x1+x2)/(n1+n2)=(264+329)/(318+362)=593/680
q̄=1-(593/680)=87/680
Transcript text: Suppose that for fast food restaurant A, it was found that 264 orders were accurate and 54 orders were not level to test the claim B, 329 orders were accurate and 33 orders were not accurate. Use a 0.05 significance laim that restaurants A \& B have the same accuracy rates. Let $P_{1}=$ prop of restaurant $A$ orders (accurate) $P_{2}=$ prop of restaurant B orders laccurate) Claim: $P_{1}=P_{2}\left(H_{0}\right)$ Counterclaim: $P_{1} \neq P_{2}\left(H_{1}\right)$ \[ \begin{array}{l} \text { "two tailed test" } \\ \bar{p}=\frac{x_{1}+x_{2}}{n_{1}+n_{2}}=\frac{264+329}{318+362}=\frac{593}{680} \\ \bar{q}=1-\frac{593}{680}=\frac{87^{585}}{680} \end{array} \]
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Solution

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Solution Steps

Step 1: Calculate Proportions

For Restaurant A, the proportion of accurate orders is given by:

\[ P_1 = \frac{264}{318} \approx 0.8302 \]

For Restaurant B, the proportion of accurate orders is:

\[ P_2 = \frac{329}{362} \approx 0.9088 \]

Step 2: Calculate Combined Proportion

The combined proportion of accurate orders across both restaurants is:

\[ \bar{p} = \frac{264 + 329}{318 + 362} = \frac{593}{680} \approx 0.8721 \]

The combined proportion of inaccurate orders is:

\[ \bar{q} = 1 - \bar{p} \approx 0.1279 \]

Step 3: Calculate Standard Error

The standard error (SE) for the difference in proportions is calculated as follows:

\[ SE = \sqrt{\frac{P_1(1 - P_1)}{n_1} + \frac{P_2(1 - P_2)}{n_2}} = \sqrt{\frac{0.141}{318} + \frac{0.0829}{362}} \approx 0.0259 \]

Step 4: Calculate Test Statistic

The test statistic \( z \) is computed using the formula:

\[ z = \frac{P_1 - P_2}{SE} = \frac{0.8302 - 0.9088}{0.0259} \approx -3.0336 \]

Step 5: Calculate P-value

The P-value corresponding to the test statistic is:

\[ P(Z > -3.0336) \approx 0.0024 \]

Step 6: Determine Critical Value

For a significance level of \( \alpha = 0.05 \) in a two-tailed test, the critical Z-value is:

\[ Z_{critical} \approx 1.96 \]

Step 7: Conclusion

Since the calculated P-value \( 0.0024 \) is less than the significance level \( 0.05 \), we reject the null hypothesis \( H_0: P_1 = P_2 \). This indicates that there is a statistically significant difference in the accuracy rates of orders between Restaurant A and Restaurant B.

Final Answer

\(\boxed{\text{Reject } H_0}\)

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