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.
a. 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.
b Construct and interpret a 99% confidence interval for the difference in the mean service times at McDonald’s and Burger King drive-thrus.
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.
a. 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.
b Construct and interpret a 99% confidence interval for the difference in the mean service times at McDonald’s and Burger King drive-thrus.
1 answer:
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Answer:
The system of equations has no solution.
Step-by-step explanation:
Given the system of equations
2y = 2x+7
x = y+2
solving the system of equations
Arrange equation variables for elimination
Multiply -y+x=2 by 2: -2y+2x=4
so adding
so the system of equations becomes
0 = 11 is false, therefore the system of equations has no solution.
Thus,
No Solution!
Answer:
a) P(12.99 ≤ X ≤ 13.01) = 0.3840
b) P(X ≥ 13.01) = 0.3075
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the cental limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
(a) Calculate P(12.99 ≤ X ≤ 13.01) when n = 16.
Here we have
This probability is the pvalue of Z when X = 13.01 subtracted by the pvalue of Z when X = 12.99.
X = 13.01
By the Central Limit Theorem
has a pvalue of 0.6915
X = 12.99
has a pvalue of 0.3075
0.6915 - 0.3075 = 0.3840
P(12.99 ≤ X ≤ 13.01) = 0.3840
(b) How likely is it that the sample mean diameter exceeds 13.01 when n = 25?
P(X ≥ 13.01) =
This is 1 subtracted by the pvalue of Z when X = 13.01. So
has a pvalue of 0.6915
1 - 0.6915 = 0.3075
P(X ≥ 13.01) = 0.3075