Let x be the average number of employees in a group health insurance plan, and let y be the average administrative cost as a percentage of claims.

x 3 7 15 39 73

y 40 35 30 25 20

(a) Make a scatter diagram of the data and visualize the line you think best fits the data.

(b) Would you say the correlation is low, moderate, or strong? positive or negative?pLEASE SELECT CORRECT ANSWER

moderate and positive

low and negative

moderate and negative

low and positive

strong and positive

strong and negative

(c) Use a calculator to verify that Σx = 137, Σx2 = 7133, Σy = 150, Σy2 = 4750, and Σxy = 3250. Compute r. (Round your answer to three decimal places.) r =

As x increases, does the value of r imply that y should tend to increase or decrease? Explain. SELECT CORRECT ANSWER

Given our value of r, y should tend to decrease as x increases.

Given our value of r, y should tend to increase as x increases.

Given our value of r, y should tend to remain constant as x increases.

Given our value of r, we cannot draw any conclusions for the behavior of y as x increases.

A multinational firm wants to estimate the average number of hours in a month that their employees spend on social media while on the job. A random sample of 83 employees showed that they spent an average of 21.5 hours per month on social media, with a standard deviation of 2.5. Construct and interpret a 95% confidence interval for the population mean hours spent on social media per month.

Exercises for Probability

Exercise 2-17

Evertight, a leading manufacturer of quality nails, produces 1-, 2-, 3-, 4-, and 5-inch nails for various uses. In the production process, if there is an over- run or the nails are slightly defective, they are placed in a common bin. Yesterday, 651 of the 1-inch nails, 243 of the 2-inch nails, 41 of the 3-inch nails, 451 of the 4-inch nails, and 333 of the 5-inch nails were placed in the bin.

1. What is the probability of reaching into the bin and getting a 4-inch nail?
2. What is the probability of getting a 5-inch nail?
3. If a particular application requires a nail that is 3 inches or shorter, what is the probability of getting a nail that will satisfy the requirements of the application?

Exercise 2-18

Last year, at Northern Manufacturing Company, 200 people had colds during the year. One hundred fifty- five people who did no exercising had colds, and the remainder of the people with colds were involved in a weekly exercise program.  Half of the 1,000 employees were involved in some type of exercise.

1. What is the probability that an employee will have a cold next year?
2. Given that an employee is involved in an exercise program, what is the probability that he or she will get a cold next year?
3. What is the probability that an employee who is not involved in an exercise program will get a cold next year?
4. Are exercising and getting a cold independent events? Explain your answer.

Exercise 2-27

In a sample of 1,000 representing a survey from the entire population, 650 people were from Laketown, and the rest of the people were from River City. Out of the sample, 19 people had some form of cancer.  Thirteen of these people were from Laketown.

1. Are the events of living in Laketown and having some sort of cancer independent?
2. Which city would you prefer to live in, assuming that your main objective was to avoid having cancer?

Exercise 1

An engineering company advertises a job in three papers, A, B and C. It is known that these papers attract undergraduate engineering readerships in the proportions 2:3:1. The probabilities that an engineering undergraduate sees and replies to the job advertisement in these papers are 0.002, 0.001 and 0.005 respectively. Assume that the undergraduate sees only one job advertisement.

1. If the engineering company receives only one reply to it advertisements, calculate the probability that the applicant has seen the job advertised in place A.
2. If the company receives two replies, what is the probability that both applicants saw the job advertised in paper A?

Exercise 2-33

Gary Schwartz is the top salesman for his company.  Records indicate that he makes a sale on 70% of his sales calls. If he calls on four potential clients, what is the probability that he makes exactly 3 sales? What is the probability that he makes exactly 4 sales?

Exercise 2

A special five football matches series will be played between UAE and KSA. The probability that UAE will win a match against KSA is 0.6.              

            (i). What is the probability that UAE will win at least 3 matches in the series?        

            (ii). What is the probability that UAE will lose all the matches of the series?          

            (iii). What is the probability that UAE will win at most 2 matches in the series?     

Exercise 2-34

Trowbridge Manufacturing produces cases for personal computers and other electronic equipment. The quality control inspector for this company believes that a particular process is out of control. Normally, only 5% of all cases are deemed defective due to   discolorations. If 6 such cases are sampled, what is the probability that there will be 0 defective cases if the process is operating correctly? What is the probability that there will be exactly 1 defective case?

Exercise 2-34

If 10% of all disk drives produced on an assembly line are defective, what is the probability that there will be exactly one defect in a random sample of 5 of these? What is the probability that there will be no defects in a random sample of 5?

Exercise 2-38

Steve Goodman, production foreman for the Florida Gold Fruit Company, estimates that the average sale of oranges is 4,700 and the standard deviation is 500 oranges. Sales follow a normal distribution.

(a)  What is the probability that sales will be greater than 5,500 oranges?

(b)  What is the probability that sales will be greater than 4,500 oranges?

(c)  What is the probability that sales will be less than 4,900 oranges?

(d) What is the probability that sales will be less than 4,300 oranges?

Exercise 2-41

The time to complete a construction project is normally distributed with a mean of 60 weeks and a standard deviation of 4 weeks.

(a)  What is the probability the project will be finished in 62 weeks or less?

(b) What is the probability the project will be finished in 66 weeks or less?

(c)  What is the probability the project will take longer than 65 weeks?

Exercise 2-42

A new integrated computer system is to be installed worldwide for a major corporation.  Bids on this project are being solicited, and the contract will be awarded to one of the bidders. As a part of the proposal for this project, bidders must specify how long the project will take. There will be a significant penalty for finishing late. One potential contractor determines that the average time to complete a project of this type is 40 weeks with a standard deviation of 5 weeks. The time required to complete this project is assumed to be normally distributed.

1. If the due date of this project is set at 40 weeks, what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?
2. If the due date of this project is set at 43 weeks what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?
3. If the bidder wishes to set the due date in the proposal so that there is only a 5% chance of being late (and consequently only a 5% chance of having to pay a penalty), what due date should be set?

Exercise 2-43

Patients arrive at the emergency room of Costa   Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.

(a)  Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4, and 5 arrivals per day.

(b)  What is the sum of these probabilities, and why is the number less than 1?

Exercise 2-44

Using the data in Problem 2-43, determine the probability of more than 3 visits for emergency room service on any given day.

Blood type AB is found in only 3% of the population.† If 330 people are chosen at random, find the probability of the following. (Use the normal approximation. Round your answers to four decimal places.)

(a) 5 or more will have this blood type


(b) between 5 and 10 will have this blood type

We are interested in determining if the amount of sleep someone gets per night affects how much they exercise. The following data are provided:

(Sleep- Exercise) 4-20, 5-0, 5.5- ,7 6 -14, 6.5- 7, 7 -12, 7.5- 4, 8- 6, 8.5- 1, 9- 8

i. Find s^2.

j. Find the test statistic for testing if b_1 is significant or not.

k. What conclusion would you make based on the test statistic found above (use .05 confidence level)?

l. Find the 95% confidence interval for β_1.

m. Assuming someone sleeps 8 hours a night, how much would you expect them to exercise?

n. Assuming someone sleeps 8 hours a night, what is the 90% confidence interval for the expected amount of exercise?

o. Assuming someone sleeps 8 hours a night, what is the 90% prediction interval for the expected amount of exercise?

The results in the form of a Minitab output are shown below. Use this output to answer the questions that follow.

------------------------------------------------------------------------------------------------------------------------------------------

Descriptive Statistics: HOURS

Variable   N         Mean        SE Mean       StDev        Minimum     Q1        Median     Q3    Maximum

HOURS     95      16.6            0.318            3.10            6.0               15.0       16.0          18.0   24.0

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1. Measures of Central Tendency: Compare the Mean to the Median. What is your conclusion regarding the shape of the distribution, degree of skewness, and the possible presence of outliers? Explain your answer in the space provided below.

2. Measures of Variation: Find the value for the Mean and the Standard Deviation in the Minitab output.
   1. Use these statistics to calculate the Coefficient of Variation. Show your calculation and final answer in the space provided below.
   2. Then interpret the value of CV. How would you use this statistic to draw a conclusion regarding the degree of variability among values in the sample?

3. Standardized Values: Assume that a randomly selected consumer reported that she spends 26 hours per week online.
   1. Calculate the equivalent Z-score for this consumer’s weekly online activity. Show your calculation and final answer in the space provided below.
   2. Using your calculation of the converted Z-score, would you conclude that this consumer is an outlier? Why or why not?

4. Empirical Rule: Assume a near normal distribution exists for this population. Answer the following questions regarding the values found in the Minitab output for this study. Show your calculations and final answer in the spaces provided.

1. 68% of all observations will fall within two values. What are these two values?

2. 95% of all observations will fall within two values. What are these two values?

3. Virtually all (over 99%) of all observations will fall within two values. What are these two values?

1. A trial evaluated the fever-inducing effects of three substances. Study subjects were adults seen in an emergency room with diagnoses of the flu and body temperatures between 100.0 and 100.9ºF. The three treatments (aspirin, ibuprofen and acetaminophen) were assigned randomly to study subjects. Body temperatures were reevaluated 2 hours after administration of treatments. The below table lists the data.

Data Table: Decreases in body temperature (degrees Fahrenheit) for each patient                            

The ANOVA table that corresponds to this data is below.

1. State the research question that this ANOVA answers.
2. Answer your research question using the means in the Data Table and the ANOVA results.
3. Which treatment(s) would you recommend to reduce a fever for this population?
4. What type of tests could you conduct that would allow you to compare each treatment group to the other (2 at a time) without inflating the type I error (α)?
5. Why is it important to make sure you do not increase the type I error?

ANOVA Table:

A psychologist is examining the educational advantages of a preschool program and suspects that there will be significant (α = .01) differences in achievement among 4th graders based on whether or not they attended preschool. Twenty-three randomly selected 4th grade children are used in the study. Twelve attended a preschool program and eleven did not (see the data below).

Group 1 - Preschool: 8, 6, 8, 9, 7, 9, 6, 9, 8, 9, 7, 8

Group 2 - No Preschool: 6, 5, 7, 6, 8, 5, 7, 5, 6, 7, 5

What is the alternative hypothesis for this independent samples t-test?

Group of answer choices µ1 - µ2 ≥ 0 µ1 - µ2 = 0 µ1 - µ2 ≠ 0 µ1 - µ2 ≤ 0 µ1 - µ2 µ1 - µ2 > 0

Golf and Chi-Square Tests

Discussion

Introduction

This discussion provides a simulated exercise using two of the most popular Chi-Square statistical tests. You are strongly encouraged to complete the textbook reading and start the MyStatLab Homework assignment before beginning this discussion. You need to be familiar with the Chi-Square distribution, its interpretation, and how results are typically calculated and reported together.

In this discussion, you are required to calculate and interpret your findings.

Review the information in each section and participate in the discussion.

Golf Rounds Scenario

As the Director of Golf for the Links Group, you are trying to determine if there is a significant difference in the number of rounds of golf played based on the day of the week. So far, you've gathered the following sample information for 520 rounds.

Now, you need to use the sample to answer these questions. For each question, the null hypothesis is that the cell categories are equal, and the significance level is .05.

• Is there a significant difference in the rounds of golf played during the week? Why is your choice appropriate?
• What is the critical value and number of degrees of freedom for the Chi-Square statistic?
• What is your decision regarding the null hypothesis? Why is your decision appropriate?

Golf Ball Quality Scenario

In addition to determining if there is a significant difference in the number of rounds of golf played based on the day of the week, you also have to determine which brand of "range balls" to buy for use at the driving range. You are looking for the most durable ball that will hold up for an extended period. You obtained samples of 100 golf balls from four different manufacturers, and your teaching professionals tested them for durability. The table shows the numbers of unacceptable and acceptable golf balls by manufacturer.

At the .05 significance level, is there a difference in the durability of the range balls? Why is your decision appropriate?

Additional Instructions

Use Excel or Statdisk to answer the questions posed in both scenarios. Remember, the focus of this discussion is understanding and interpreting two different Chi-Square tests.

6. A calculated correlation coefficient of 0.99 between variables A and B implies that variable A causes variable B.

7. Which of the following is NOT a possible value for a calculated F statistic?

8. A large student organization at CSUN claims it has an equal distribution of people from each of the 9 colleges (e.g., business, humanities). You are interested in investigating whether or not this claim is accurate. To do so, you first take a sample of students from the organization. What would be the most appropriate test to utilize with your sample data to help answer this research question?

9. You are presented with 4 different correlation coefficients (r): 0.65, -0.34, -0.86, and 0.19. Which of the following lists correctly represents the 4 r’s placed in order of strength from weakest to strongest.

10.

In which direction is the χ² (chi-square) distribution skewed?

The National Collegiate Athletic Association (NCAA) requires Division II athletes to score at least 820 on the combined mathematics and reading parts of the SAT in order to compete in their first college year. The scores of the 1.5 million high school seniors taking the SAT last year are approximately Normal with mean 1026 and standard deviation 209. For an SRS size 200

1) Find the mean and standard deviation of x bar

2) What is the distribution of x bar?

3) What is the probability that a sample mean value exceeds 1028?

4) The highest 2.5% of sample mean value are higher than ____

5) Has the average score increased since last year? To answer this, do the followings:

I. A SRS size 200 gives a sample mean of 1028. State hypotheses, find the test statistic, pvalue, and express your conclusion using a significance level of α=2.5%.

a. Hypotheses (Use notations)

b. test statistic

c. pvalue

d. Conclusion? Include practical terms.

II. Find a 95% confidence interval for the mean SAT score.

A company uses a combination of three components- A, B and C to create three different drone designs. The first design Glider uses 3 parts of component A and 2 parts of components B. Design Blimp uses 2 parts of component B and C, and the last design, Pilot uses one part of each component. A sample of 75 components, 25 A, 25 B, 25 C, will be used to make prototypes for the various designs. If 30 components are selected at random, what is the likelihood two prototypes of each design can be made?

You may need to use the appropriate technology to answer this question.

An amusement park studied methods for decreasing the waiting time (minutes) for rides by loading and unloading riders more efficiently. Two alternative loading/unloading methods have been proposed. To account for potential differences due to the type of ride and the possible interaction between the method of loading and unloading and the type of ride, a factorial experiment was designed. Use the following data to test for any significant effect due to the loading and unloading method, the type of ride, and interaction. Use α = 0.05.

Find the value of the test statistic for method of loading and unloading.

Find the p-value for method of loading and unloading. (Round your answer to three decimal places.)

p-value =

State your conclusion about method of loading and unloading.

Because the p-value > α = 0.05, method of loading and unloading is not significant.Because the p-value ≤ α = 0.05, method of loading and unloading is significant.     Because the p-value ≤ α = 0.05, method of loading and unloading is not significant.Because the p-value > α = 0.05, method of loading and unloading is significant.

Find the value of the test statistic for type of ride.

Find the p-value for type of ride. (Round your answer to three decimal places.)

p-value =

State your conclusion about type of ride.

Because the p-value ≤ α = 0.05, type of ride is not significant.Because the p-value ≤ α = 0.05, type of ride is significant.     Because the p-value > α = 0.05, type of ride is not significant.Because the p-value > α = 0.05, type of ride is significant.

Find the value of the test statistic for interaction between method of loading and unloading and type of ride.

Find the p-value for interaction between method of loading and unloading and type of ride. (Round your answer to three decimal places.)

p-value =

State your conclusion about interaction between method of loading and unloading and type of ride.

Because the p-value > α = 0.05, interaction between method of loading and unloading and type of ride is significant.Because the p-value > α = 0.05, interaction between method of loading and unloading and type of ride is not significant.     Because the p-value ≤ α = 0.05, interaction between method of loading and unloading and type of ride is significant.Because the p-value ≤ α = 0.05, interaction between method of loading and unloading and type of ride is not significant.

2. You may need to use the appropriate technology to answer this question.

The calculations for a factorial experiment involving four levels of factor A, three levels of factor B, and three replications resulted in the following data: SST = 282, SSA = 26, SSB = 22, SSAB = 179.Set up the ANOVA table. (Round your values for mean squares and F to two decimal places, and your p-values to three decimal places.)

Test for any significant main effects and any interaction effect. Use α = 0.05.

Find the value of the test statistic for factor A. (Round your answer to two decimal places.)

Find the p-value for factor A. (Round your answer to three decimal places.)

p-value =

State your conclusion about factor A.

Because the p-value ≤ α = 0.05, factor A is not significant.Because the p-value ≤ α = 0.05, factor A is significant.     Because the p-value > α = 0.05, factor A is not significant.Because the p-value > α = 0.05, factor A is significant.

Find the value of the test statistic for factor B. (Round your answer to two decimal places.)

Find the p-value for factor B. (Round your answer to three decimal places.)

p-value =

State your conclusion about factor B.

Because the p-value ≤ α = 0.05, factor B is significant.Because the p-value ≤ α = 0.05, factor B is not significant.     Because the p-value > α = 0.05, factor B is not significant.Because the p-value > α = 0.05, factor B is significant.

Find the value of the test statistic for the interaction between factors A and B. (Round your answer to two decimal places.)

Find the p-value for the interaction between factors A and B. (Round your answer to three decimal places.)

p-value =

State your conclusion about the interaction between factors A and B.

Because the p-value > α = 0.05, the interaction between factors A and B is not significant.Because the p-value ≤ α = 0.05, the interaction between factors A and B is not significant.     Because the p-value ≤ α = 0.05, the interaction between factors A and B is significant.Because the p-value > α = 0.05, the interaction between factors A and B is significant.

Find the critical value of t for each of the following:

Please show work and explain.

a. 1-α = 0.95, n=21

b. 1-α = 0.99, n=21

c. 1-α = 0.90, n=32

d. 1-α = 0.99, n=65