STATE: Which type of fast‑food chain fills orders most accurately at the drive‑thru window? The Quick Service Restaurant (QSR) magazine drive‑thru study visits restaurants in the largest fast‑food chains in all 50 states. Visits occurred throughout the day, starting at 11:00 A. M and ending at 2:30 A. M . During each visit, the researcher ordered a main item, a side item, and a drink and made a minor special request such as a beverage with no ice. After receiving the order, all food and drink items were checked for complete accuracy. Any food or drink item received that was not exactly as ordered resulted in the order being classified as inaccurate. Also included in the measurement of accuracy were condiments asked for, napkins, straws, and correct change. Any errors in these resulted in the order being classified as innacurate. In 2015, the ethnic category including Del Taco, El Pollo Loco, Fazoli's, Panda Express, Taco Bell, and Taco John's had the fewest inaccuracies, with only 42 of 457 orders classified as inaccurate. What proportion of orders are filled accurately in the ethnic fast‑food category? Use 95% confidence.

See explaination

Explanation:

We are given that : In 2015, the ethnic category including Del Taco, El Pollo Loco, Fazoli's, Panda Express, Taco Bell, and Taco John's had the fewest inaccuracies, with only 42 of 457 orders classified as inaccurate.

Thus x = number of orders that are filled accurately in the ethnic fast?food category = 457 - 42 = 415

Thus sample proportion of orders are filled accurately in the ethnic fast?food category is :

\hat{p}=\frac{x}{n}

\hat{p}=\frac{415}{457}

\hat{p}=0.9081

Also we have to find 95% confidence interval for the population proportion of orders that are filled accurately in the ethnic fast?food category.

Formula:

( \hat{p} - E \: \: ,\: \: \hat{p} + E )

where

E = Z_{c}\times \sqrt{\frac{\hat{p}\times (1-\hat{p})}{n}}

find Zc value for c =95% confidence level.

Find Area = ( 1 + c) / 2 = ( 1 + 0.95) / 2 = 1.95/2 = 0.9750

Look in z table for area = 0.9750 or its closest area and find corresponding z value.

See attached file for z table

Thus Zc = 1.96

E = Z_{c}\times \sqrt{\frac{\hat{p}\times (1-\hat{p})}{n}}

E = 1.96 \times \sqrt{\frac{0.9081 \times (1-0.9081 )}{457}}

E = 1.96 \times \sqrt{\frac{0.9081 \times 0.0919 }{457}}

E = 1.96 \times \sqrt{0.000183 }

E = 1.96 \times 0.013514

E =0.026487

E =0.0265

Thus we get :

( \hat{p} - E \: \: ,\: \: \hat{p} + E )

( 0.9081 - 0.0265 \: \: ,\: \: 0.9081 + 0.0265 )

( 0.8816 \: \: ,\: \: \: 0.9346 )

Thus a 95% confidence interval for the population proportion of orders that are filled accurately in the ethnic fast?food category is in between ( 0.8816 \: \: ,\: \: \: 0.9346 ) .

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