The following counts of vehicles arriving at a toll station, over 1-minute intervals, were made by an engineering student: 2,0,4,1,2,4,2,1,6,2,5,4,4,3,3,2,2,1,2,3,2,5,2,0,1,2,0,3,1,1, 2,3,5,6,3,4,3,4,0,6,3,2,1,4,4,0,1,4,3,7,0,0,2,3,2,4,3,2,4,5 (a) Assuming the vehicle arrivals were generated by a Poisson process, compute the maximum likelihood estimate of the arrival rate. (b) Test the hypothesis that the arrival counts follow a Poisson distribution, at the α=0.05 significance level.

Drive-through Service Time at McDonald’s When you are on the go and looking for a quick meal, where do you go? If you are like millions of people every day, you make a stop at McDonald’s. Known as “quick service restaurants” in the industry (not “fast food”), companies such as McDonald’s invest heavily to determine the most efficient and effective ways to provide fast, high quality service in all phases of their business. Drive-through operations play a vital role. It’s not surprising that attention is focused on the drive-through process. After all, over 60% of the individual restaurant revenues in the United States come from the drive-through operations. Yet understanding the process is more complex than just counting cars. Marla King, professor at the company’s international training center, Hamburger University, got her start 25 years ago working at a McDonald’s drive-through. She now coaches new restaurant owners and managers. “Our stated drive-through service time is 90 seconds or less. We train every manager and team member to understand that a quality customer experience at the drive-through depends on them,” says Marla. Some of the factors that affect a customers’ ability to complete their purchases with 90 seconds include restaurant staffing, equipment layout in the restaurant, training, and efficiency of the grill team, and frequency of customer arrivals to name a few. Customer order patterns also play a role. Some customers will just order drinks, while others seem to need enough food to feed an entire soccer team. And then there are the special orders. Obviously, there is plenty of room for variability here. Yet that doesn’t stop the company from using statistical techniques to better understand the drive-through action. In particular, McDonald’s utilizes numerical measures of the center (mean) and spread (variance) in the data and to help transform the data into useful information. In order for restaurant managers to achieve the goal in their own restaurants, they need training in proper restaurant and drive-through operations. Hamburger University, McDonald’s training center located near Chicago, Illinois, satisfies that need. In the mock-up restaurant service lab, managers go through a “before and after” training scenario. In the “before” scenario, they run the restaurant for thirty minutes as if they were back in their home restaurants. Managers in the training class are assigned to be crew, customers, drive-through cars, special needs guests (such as hearing impaired), or observers. Statistical data about the operations, revenues, and service times are collected and analyzed. Without the right training, the restaurant’s operation usually starts breaking down after 10-15 minutes. After debriefing and analyzing the data collected, the managers make suggestions for adjustments and head back to the service lab to try again. This time, the results usually come in well within standards. “When presented with the quantitative results, managers are pretty quick to make the connections between better operations, higher revenues, and happier customers,” Marla states. When managers return to their respective restaurants, the training results and techniques are shared with staff who are charged with implementing the ideas locally. The results of the training eventually are measured when McDonald’s conducts a restaurant operations improvement process study, or ROIP. The goal is simple: improved operations. When the ROIP review is completed, statistical analyses are performed and managers are given their results. Depending on the results, decisions might be made that require additional financial resources, building construction, staff training, or reconfiguring layouts. Yet one thing is clear: Statistics drive the decisions behind McDonald’s drive-through service operations.

Questions:

1. After returning from the training session at Hamburger University, a McDonald’s store owner
selected a random sample of 323 drive-through customers and carefully measured the time it took
from when a customer entered the McDonald’s property until the customer had received the order at
the drive-through window.
These data are provided, using Excel spreadsheet. Note that the owner
selected some customers during the breakfast period, others during lunch or dinner time. For the
overall sample, compute the key measures of the central tendency and variation.
Based on these measures, what conclusion might the owner reach with respect to how well his store is
doing in meeting the 90 second customer service goal? Support your argument with appropriate
hypothesis testing.

2. Compute the key measures of central tendency and variation for drive-through times broken down by
breakfast, lunch, and dinner time periods. Based on these calculations, does it appear that the store is
doing better at one of these time periods than the others in providing shorter drive-through waiting
times? Support your argument with appropriate hypothesis testing.

3. Determine if there are any outliers in the sample data. Discuss.

show the steps of doing it in excel when answering it please

Share a link to a news article or study that you think exhibits a sampling error or bias. Briefly describe what you think the issue is and how you might fix it.

The factory building manager at Delectable Delights, Jason Short, is concerned that the new contractor he hired is taking too long to replace defective lights in the factory workspace. He would like to perform a hypothesis test to determine if the replacement time for the lights under the new contractor is in fact longer than the replacement time under the previous contractor, which was 3.2 days on average. He selects a random sample of 12 service calls to replace defective lights and obtains the following times to replacement (in days). Use a significance level of 0.05.

6.2       7.1       5.4       5.5       7.5       2.6       4.3       2.9       3.7       0.7       5.6       1.7

Define μ in the context of the problem and state the appropriate hypotheses. (5 pts)

Regardless of your results in Part B, calculate the appropriate test statistic by hand. Write out all your steps. (5 pts)

1. What is your decision regarding the null hypothesis? (In other words, do you reject or fail to reject and why?) (5 pts)
   1. 
      

Write a final concluding statement to Jason giving the results of the hypothesis test. (In other words, write the final summary statement.) (5 pts)

In the following table, the random variable x represents the number of laptop computers that failed during a drop-test of six sample laptops. Use the table to answer the questions a) - e) below.
x 0 1 2 3 4 5 6

P(x) 0.377 0.399 0.176 0.041 0.005 0.000 0.000

a) Find and report the mean and the standard deviation of this distribution.

b) Using the range rule of thumb, identify the range of values containing the usual number of laptop failures among the six laptops that were tested. Is three laptops an unusually high number of failures among six tested? Explain.

c) Find the probability of getting exactly one laptop that fails among six laptops tested.

d) Find the probability of getting one or fewer laptops that fail among six laptops tested.

e) Which probability is most relevant for determining whether one laptop is an unusually low number of laptops to fail among six laptops tested: the result from part (c) or (d)? f) Is one laptop an unusually low number of laptops that fail among six laptops tested? Why or why not?

1) In a recent poll, the Gallup organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. Answer the questions below, showing work. Bare answers are not acceptable. (Showing work means writing the calculator command you used with correct input values in the correct order.)

f) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values in the United States is poor? Why or why not?

g) Now based on a random sample of 500 adult Americans, compute the mean and standard deviation of the random variable X, the number of adults who believe the overall state of moral values in the United States is poor. Interpret the mean.

h) Would it be unusual to identify 240 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans? Why? Now using the Normal Approximation to the Binomial, approximate the probability that:

i) Exactly 250 of those surveyed feel the overall state of moral values in the United States is poor.

j) Between 220 and 250, inclusive, of those surveyed feel the overall state of moral values in the United States is poor.

k) At least 260 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans. Would this be unusual? Why?

Bayus (1991) studied the mean numbers of auto dealers visited by early and late replacement buyers. Letting μ be the mean number of dealers visited by all late replacement buyers, set up the null and alternative hypotheses needed if we wish to attempt to provide evidence that μ differs from 4 dealers. A random sample of 100 late replacement buyers yields a mean and a standard deviation of the number of dealers visited of x⎯⎯x¯ = 4.26 and s = .52. Using a critical value and assuming approximate normality to test the hypotheses you set up by setting α equal to .10, .05, .01, and .001. Do we estimate that μ is less than 4 or greater than 4? (Round your answers to 3 decimal places.)

H₀ : μ (Click to select)=≠ 4 versus H_a : μ (Click to select)≠= 4.

t         

There is (Click to select)noextremely strongvery strongstrongweak evidence.

μ is (Click to select)less thangreater than 4.

Suppose a random sample of 10,000 individuals is asked to identify their favorite brand of soap among ten choices. The following results from the survey are obtained:

Observed
Brand Frequency

A             1200
B             900
C             850
D             1160
E                1020
F               975
G            1100
H            980
I              1035
J               780

Test the hypothesis that the preferences for each brand are equal (or uniform). Test this at the 0.05 level.

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 63 professional actors, it was found that 41 were extroverts.

(a) Let p represent the proportion of all actors who are extroverts. Find point estimates for p and q. (Round your answer to four decimal places.)

p̂=

q̂=

(b) Find a 95% confidence interval for p. (Round your answers to two decimal places.)

Find the maximal margin of error. (Round your answer to two decimal places.)

E =

Report the bounds from the 95% confidence interval for p. (Round your answers to two decimal places.)

lower limit =

upper limit =

We need to find the confidence interval for the SHOE SIZE variable. To do this, we need to find the mean and standard deviation with the Week 1 spreadsheet. Then we can the Week 5 spreadsheet to find the confidence interval. This does not need to be separated by males and females, rather one interval for the entire data set.

First, find the mean and standard deviation by copying the SHOE SIZE variable and pasting it into the Week 1 spreadsheet. Write down the mean and the sample standard deviation as well as the count. Open the Week 5 spreadsheet and type in the values needed in the green cells at the top to find the confidence interval.

1. Give and interpret the 95% confidence interval for the size of students’ shoes.

Change the confidence level to 99% to find the 99% confidence interval for the SHOE SIZE variable.

2. Give and interpret the 99% confidence interval for the size of students’ shoes.

3. Compare the 95% and 99% confidence intervals for the size of students’ shoes. Explain the difference between these intervals and why this difference occurs.

1a)Explain when you can approximate a hypogeometric distribution using a binomial distribution. Why can we do this? Use an example to illustrate the approximation.

b) Prove that if X follows a uniform distribution, the expectation is the average of all the outcomes.

1. How many scores are between 2 standard deviations above and below the mean in a normal distribution?
2. A score is 3 standard deviations above the mean in a normal distribution. How much of the data is below that score?
3. How many combinations can nine pair of shoes be made for nine people?

For the following questions, find the probability using a standard 52-card deck. Write your answer as a fraction or with a colon in lowest terms.

1. Find the probability of drawing a face card.
2. Find the probability of drawing a black card.
3. Find the probability of drawing a red 10.
4. Find the probability of drawing a king.
5. Find the probability of not drawing a face card.
6. Find the probability of drawing a number less than 4.
7. Find the probability of drawing a 5.
8. Find the probability of drawing a black 2.
9. Find the probability of not drawing a red 8.
10. Find the probability of drawing an ace of spades.

Complete the frequency table above by filling in the frequency of raw scores occurring in each interval, using the following data:

30, 32, 11, 14, 40, 37, 16, 26, 12, 33, 13, 19, 38, 12, 28, 15, 39, 11, 37, 17, 27, 14, 36

Interval 11 - 15:

Interval 16 - 20:

Interval 21 - 25:

Interval 26 - 30:

Interval 31 - 35:

Interval 36 - 40:

Montarello and Martins (2005) found that fifth grade students completed more mathematics problems correctly when simple problems were mixed in with their regular math assignments. To further explore this phenomenon, suppose that a researcher selects a standardized mathematics achievement test that produces a normal distribution of scores with a mean of µ= 100 and a standard deviation of σ = 18. The researcher modifies the test by inserting a set of very easy problems among the standardized questions and gives the modified test to a sample of n = 36 students. If the average test score for the sample is M = 104, is this result sufficient to conclude that inserting the easy questions improves student performance? Use a one-tailed test with α = .01.

A)The alternative hypotheses in words is

B)The null hypothesis in symbols is

C)The critical z values is

D)The z-score statistic is:

E) Your decision is