Question

More than $70 billion is spent each year in the drive-thru lanes of America’s fast-food restaurants. Having quick, accurate, and friendly service at a drive-thru window translates directly into revenue for the restaurant. According to Jack Greenberg, former CEO of McDonald’s, sales increase 1% for every six seconds saved at the drive-thru. So industry executives, stockholders, and analysts closely follow the ratings of fast-food drive-thru lanes that appear annually in QSR, a publication that reports on the quick-service restaurant industry.

The 2012 QSR magazine drive-thru study involved visits to a random sample of restaurants in the 20 largest fast-food chains in all 50 states. During each visit, the researcher ordered a modified main item (for example, a hamburger with no pickles), a side item, and a drink. If any item was not received as ordered, or if the restaurant failed to give the correct change or supply a straw and a napkin, then the order was considered “inaccurate.” Service time, which is the time from when the car stopped at the speaker to when the entire order was received, was measured each visit. Researchers also recorded whether or not
each restaurant had an order-confirmation board in its drive-thru.

Here are some results from the 2012 QSR study:

• For restaurants with order-confirmation boards, 1169 of 1327 visits (88.1%) resulted in accurate orders. For restaurants with no order-confirmation board, 655 of 726 visits (90.2%) resulted in accurate orders.
• McDonald’s average service time for 362 drive-thru visits was 188.83 seconds with a standard
  deviation of 17.38 seconds. Burger King’s service time for 318 drive-thru visits had a mean of 201.33
  seconds and a standard deviation of 18.85 seconds.

1. Is there a significant difference in order accuracy between restaurants with and without order-confirmation boards? Carry out an appropriate test at the a = 0.05 level to help answer this question.
2. A 95% confidence interval for the difference in the population proportions of accurate orders at restaurants with and without order-confirmation boards is (–0.049, 0.00649). Interpret the confidence interval and explain how the confidence interval is consistent with your conclusion from Question 1.

Answer 1

Hypothesis to be tested

VS

given p1=0.881 p2=0.902 n1=1327 n2=726

The test statistic where q=1-p

= -1.46

we reject null if at 5% level of significance

also the p-value calculated is 0.14 which is greater than 0.05(alpha) thus strongly accepting the null hypothesis

1.46 < 1.96, thus we do not reject the null and accept the null and conclude that there is no significant difference between the population proportions

we can see that the confidence interval is quite narrow and indicates stability with the results. The estimate is stable and indicates the range of difference which is very small (below 0.05) indicating little or no difference. The confidence interval gives us the range of possible values of the difference between the two population proportions. 95% of the time the difference in population estimate will lie in the interval -0.049 to 0.00649 thus proving the null hypothesis to be true that there is no significant difference between the two population proportions.