Exercise 1: A fair six-sided die is tossed until a 6 is observed. Let X be the number of tosses until (and including) the first 6 is observed.

a) Find the probability that at least 5 rolls are required to obtain the first 6

b) Given that a 6 was not rolled on the first 3 rolls, find the probability that at least 5 more rolls are needed to obtain the first 6.

c) Your two answers should be equal. What property of the geometric random variable is illustrated in this exercise?

Exercise 2: Consider a student taking a multiple-choice test. On a given question, either the student knows the answer, in which case he answers it correctly, or he does not know the answer, in which case he guesses hoping to guess the correct answer. Assume there are five multiple-choice alternatives. Let p be the probability that the student

knows the correct answer. Let us assume that the probability that the student gets the correct answer given that he guesses is 1/5. Show that the probability that the student 

knowsthe correct answer given that the student gotthe correct answer is 5p/ 4p+1

HINTUse the Theorem of Total Probability together with Bayes’ Theorem. Carefully define the events.

Exercise 3: In this game, there are five fair six-sided die. Begin by rolling all five dice (round 1). The objective is to obtain five 6s in exactly two rounds. For example, if two 6s were rolled in the first round, then you roll the remaining three dice in the second round in the hopes of rolling three 6s. If only one 6 was rolled in the first round, you roll the remaining four dice in the second round in the hopes of rolling four 6s. If no 6s are rolled in the first round, then all five dice are rolled in the second round in the hopes of rolling five 6s. And if five 6s are rolled in the first round, the game is over! But you don’t win because it did not take you exactly two rounds to obtain the five 6s (and you are not allowed to try again). Use the Theorem of Total Probability to find the probability of rolling five 6s in exactly two rounds. You will make good use of the binomial probability distribution.

 

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Answer all questions. Shade the correct answer in your scantron for each of the questions on the quiz. 1.True or False: A list of the names of all students registered for classes at North Campus for the fall semester 2019 is an example of a population.(a)True(b) False2.What is the level of measurement of data consisting of the book categories of science, literature, mathematics, and history?(a)Nominal(b) Ordinal(c) Interval(d) RatioThe computer giant IBM has 329,373 employees and 637,133 stockholders. A vice-president plans to conduct a survey to study the numbers of shares held by individual stockholders.3.Are the numbers of shares held by stockholders discrete or continuous?(a)Discrete(b) Continuous4.What is the level of measurement for the number of shares held by stockholders?(a)Nominal(b) Ordinal(c) Interval(d) Ratio5.If the survey is conducted by telephoning 50 randomly selected stockholders in each of the 50 Unites States, what type of sampling is being used?(a)Random(b) Systematic(c) Convenience(d) Stratified(e) Cluster6.If a complete list of all stockholders is complied and every 500th name is selected for the survey, what type of sampling is being used?(a)Random(b) Systematic(c) Convenience(d) Stratified(e) Cluster7.If, at the annual stockholders’ meeting, a survey is conducted of all who attend, what type of sampling is being used?(a)Random(b) Systematic(c) Convenience(d) Stratified(e) Cluster8.Data that can be classified according to color are measured on what scale?(a) Nominal(b) Ordinal(c) Interval(d) Ratio 9.In an experiment when the effects from two or more variables cannot be distinguished from each other, what is said to have occurred?(a) Contrasting(b) Blinding(c) Commingling (d) Confounding  10.In the largest public health experiment ever conducted, 200,745 children were given a treatment consisting of the Salk vaccine, while 201,229 other children were given a placebo.The purpose of the experiment was to determine whether the Salk vaccine was effective in combating polio. What was the independent variable in the study?(a)whether the subject received the polio vaccine or not(b)whether the subject contracted polio or not(c)whether the subject was given a placebo or not(d)whether the subject was a male or female child(e)The salk vaccine11.The difference between a sample result and the true population result is knows as :(a) Population Error(b) Sampling Error(c) Non-sampling Error(d) Test Error12.Data that provide information about relative comparisons, but not the magnitude of the differences is known as:(a)Nominal (b) Ordinal(c) Interval(d) Ratio13.For a study in which subjects are treated with a new drug and then observed, is the study observational or is it an experiment?(a) Observational(b) Experiment14.All of the following are ways in which statistics can be misused EXCEPT:(a)Precise Numbers(b) Small Samples(c) Random Samples(d) Order of Questions15.In a ___sample, members from the population are selected in such a way that each individual member in the population has an equal chance of being selected.(a)Systematic(b)Precise(c) Stratified(d) Random16.In a ___ study, data are observed, measured, and collected at one point in time.(a)Prospective(b) Retrospective(c)Rigorous(d) Cross-Sectional17.Data in which differences are meaningful, but there is no natural zero starting point is known as:(a)Nominal(b) Ordinal(c) Interval(d) Ratio18.On the day of the last presidential election, ABC News organized an exit poll in which specific polling stations were randomly selected and all the voters were surveyed as they left the premises. What type of sampling is being used?(a)Random(b) Systematic(c) Convenience(d) Cluster19.Determine whether the given value is a statistic or parameter: The mean age of students at Broward College is 24.7 years.(a)Statistic(b) Parameter 20.Determine which of the four levels of measurement (nominal, ordinal, interval and ratio)is most appropriate for the data below:The social security numbers of students in this class(a)Nominal(b) Ordinal(c) Interval(d) RatioQuestions 21-23, pertain to the statement below:The state of Florida wanted to know what percent of adults over the age of 35, living in Florida, have a gun in their home. They used a Gallup poll in which 2059 randomly selected adults over the age of 35 living in Florida were asked the question “Do you have a gun in your home?” Results showed that 39% answered yes to the question. 21.What is the sample?(a)39% of 2059 randomly selected adults(b) Adults over the age of 35(c)2059 randomly selected adults (d) 2059 randomly selected adults over the age of 3522.What is the population?(a)39% of 2059 randomly selected adults(b) All adults over the age of 35(c)2059 randomly selected adults (d) 2059 randomly selected adults over the age of 3523.The results of the survey revealed that 39% of respondents have a gun in their home. The value of 39% is a ___?(a)Statistic(b) Parameter24.A statistics professor obtains a sample of students by selecting the first 10 students entering his classroom. What type of sampling is he using?(a)Random(b) Systematic(c) Stratified(d) Cluster(e) Convenience25.True or False. The interval level of measurement is like the ordinal level with the additional property that the difference between any two data values is meaningful. Moreover, data at this level have a natural zero starting point. (a)True(b) false 26.Which of the following statements are true?I.All variables can be classified as quantitative or qualitative variables.II.Qualitative variables can be continuous variables.III.Quantitative variables can be discrete variables. (a)I Only(b) I and III Only(c) III Only(d) I and II Only(e) All correct27.Most psychology texts report that the average IQ score for an adult male is between 90 and 110.This average is an example of a:(a)Parameter(b) Sample(c) statistic(d) population(e) None of the above28.The variable that changes due to the manipulation of the independent variable is called the:(a)Explanatory variable(b) outcome variable(c) manipulated variable 29.A sample in which the respondents themselves decide whether to be included is called a:(a)Self-serving (b)voluntary response(c) selection(d)convenience30.What is the level of measurement for academic letter grades such as A, B, C D, F or W? (a)ratio(b) ordinal(c) nominal(d) interval(e)categorical31. Questions 31-33 pertain to the following study: Agricultural researchers are investigating whether a new form of pesticide will lead to lower levels of insect damage to crops than the traditional pesticide. What is the independent variable in this study?(a) Agricultural researchers(b) Type of pesticide used(c) Levels of insect damage to crops(d) Traditional pesticide(e) Amount of crop produced32.What is the dependent variable in this study?(a) Agricultural researchers(b) Type of pesticide used(c) Levels of insect damage to crops(d) Traditional pesticide(e) Amount of crop produced33.Would this be an observational study or an experimental study?(a)Observational (b) ExperimentalQuestions 34 – 38 pertain to the following: Of the adult U.S. population, 36% has an allergy. A sample of 1200 randomly selected adults resulted in 33.2% reporting an allergy.34.What is the population?(a)All adults in the U.S. suffering from allergies(b)All adults in the U.S(c)1200 randomly selected adults(d)36% of the adult population(e)All adults in the U.S. without allergies35.What is the sample?(a)1200 randomly selected adults with allergies(b)All adults in the U.S(c)1200 randomly selected adults(d)36% of the adult population(e)33.2% reporting an allergy36.What is the value of the statistic?(a)1200 randomly selected adults(b)36%(c)33.2%(d)36% of 1200 randomly selected adults(e)33.2% of the adult population37.What is the value of the parameter?(a)33.2%(b)36% (c)1200 randomly selected adults(d)33.2% of the adult population(e)All adults in the U.S population38.Indicate which level of measurement is being used in the scenario below:A professor records the scores that students obtained on their mid-term exam.(a)Nominal(b) Ordinal(c) interval(d) ratio(e) Mid-term exam scores 39.What is another te rm for a qualitative variable?(a)numerical (b) statistical(c) categorical (d) dependent(e) experimental40.You would like to estimate the percent of students at Broward College who believe that marijuana should be legalized. You get a list of the names and addresses of all students enrolled at the college from the registrar’s office. You send a survey in the mail to all students asking whether they believe that marijuana should be legalized. Identify the problem with this study.(a)The sample is too large(b)The sample is too small(c)The survey has a loaded question(d) It is a voluntary response sample(e) The sample size is not given41.On a test, 74% of the questions are answered correctly. If 111 questions are correct, how many questions are on the test? A) 137 B) 67 C) 150 D) 74E) 8242.A gardener has 75 clients, 45% of whom are businesses. Find the number of business clients. A) 41 clients B) 34 clients C) 36 clients D) 73 clientsE) 60 clients

 

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 Write a 500-word essay explaining the concept of “members as leaders.” Using the diagram from the Unit Lesson of the key elements, apply those concepts to a healthcare team you have been part of. How did you see each element impacting the overall productivity of the team? In your essay, break down each element into a separate section of the essay. Be sure to include an introductory paragraph, several paragraphs in the body of the essay, and a solid concluding paragraph followed by a list of your references. Access the an online library, and use at least one outside source plus our textbook to support your ideas and thoughts. Use APA style for in-text citations and your reference listings. 

Weiss, D., Tilin, F., & Morgan, M. (2018). The interprofessional health care team: Leadership and development (2nd ed.). Burlington, MA: Jones & Bartlett Learning. 

 

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DATA FILE 5

ASSIGNMENT 5

1. Independent random samples of 64 observations each are chosen from two normal populations with the following means and standard deviations:

Population 1	Population 2
µ1 = 12	µ2 = 10
Ϭ1 = 4	Ϭ2 = 3

Let ₁ and ₂ denote the two sample means.

1. Give the mean and standard deviation of the sampling distribution of ₁. 

• Give the mean and standard deviation of the sampling distribution of ₂. 

• Suppose you were to calculate the difference (₁ – ₂) between the sample means.  Find the mean and standard deviation of the sampling distribution of (₁ – ₂). 

• Will the statistic (₁ – ₂) be normally distributed? Explain.  

• In order to compare the means of two populations, independent random samples of 400 observations are selected from each population, with the following results: 

Sample 1	Sample 2
1= 5,275	2 = 5,240
s1 = 150	s2 = 200

1. Use a 95% confidence interval to estimate the difference between the population means (µ₁ – µ₂).  Interpret the confidence interval.  
   1. Test the null hypothesis H₀: (µ₁ – µ₂) = 0 versus the alternative hypothesis H_a: (µ₁ – µ₂) ≠ 0.  Give the significance level of the test, and interpret the result.  
   1. Suppose the test in part b was conducted with the alternative hypothesis H_a: (µ₁ – µ₂) > 0. How would your answer to part b change? 
   1. Test the null hypothesis H₀: (µ₁ – µ₂) = 25 versus the alternative hypothesis H_a: (µ₁ – µ₂) ≠ 25.  Give the significance level and interpret the result.  Compare your answer to the test conducted in part b. 
   1. What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a – d?.

• Assume that Ϭ₁² = Ϭ₂² = Ϭ².  Calculate the pooled estimator of Ϭ² for each of the following cases:
  • s₁² = 120, s₂² = 100, n₁ = n₂ = 25
  • s₁² = 12, s₂² = 20, n₁ =20, n₂ = 10
  • s₁² = 0.15, s₂² = 0.20, n₁ =6, n₂ = 10
  • s₁² = 3,000, s₂² =2,500, n₁ =16, n₂ = 17
  • Note that the pooled estimate is a weighted average of the sample variances.  To which of the variances does the pooled estimate fall nearer in each of the above cases?

• Suppose you manage a plant that purifies its liquid waste and discharges the water into a local river.  An EPA inspector has collected water specimens of the discharge of your plant and also water specimens in the river upstream from your plant.  Each water specimen is divided into five parts, the bacteria count is read on each, and the mean count for each specimen is reported.  The average bacteria count for each of six specimens are reported in the following table for the two locations.

Plant Discharge	Upstream
30.1	36.2	33.4	29.7	30.3	26.4
28.2	29.8	34.9	27.3	31.7	32.3

1. Why might the bacteria counts shown here tend to be approximately normally distributed?  
   1. What are the appropriate null and alternative hypotheses to test whether the mean bacteria count for the plant discharge exceeds that for the upstream location? Be sure to define any symbols you use.
   1. What assumptions are necessary to ensure the validity of this test?

• A paired difference experiment produced the following data:

n_D = 18                  ₁ = 92                   ₂ = 95.5               _D = -3.5               s_D² = 21

1. Determine the values of t for which the null hypothesis, µ₁ – µ₂ = 0, would be rejected in favor of the alternative hypotheses, µ₁ – µ₂ < 0.  Use α = .10.
   1. Conduct the paired difference test described in part a.  Draw the appropriate conclusions.
   1. What assumptions are necessary so that the paired difference test will be valid?
   1. Find a 90% confidence interval for the mean difference µ_D. 
   1. Which of the two inferential procedures, the confidence interval of part d of the test of hypothesis of part b, provides more information about the differences between the population means? 

• Facility layout and material flow path design are major factors in the productivity analysis of automated manufacturing systems.  Facility layout is concerned with the location arrangement o machines and buffers for work-in-process.  Flow path design is concerned with the location arrangement of machines and buffers for work -in-process.  Flow path design is concerned with the direction of manufacturing material flows (e.g., unidirectional or bidirectional).  A manufacturer of printed circuit boards (PCBs) is interested in evaluating two alternative existing layout and flow path designs.  The output of each design was monitored for eight consecutive working days.

Working Days	Design 1	Design 2
8/16	1,220 units	1,273 units
8/17	1,092 units	1,363 units
8/18	1,136 units	1.342 units
8/19	1,205 units	1,471 units
8/20	1,086 units	1,299 units
8/23	1,274 units	1,457 units
8/24	1,145 units	1,263 units
8/25	1,281 units	1,368 units

1. Construct a 95% confidence interval for the difference in mean daily output of the two designs.               
   1. What assumptions must hold to ensure the validity of the confidence interval? 
   1. Design 2 appears to be superior to Design 1.  Is this confirmed by the confidence interval?

• Construct a 95% confidence interval for (p₁ – p₂) in each of the following situations:

1. n₁ = 400, = 0.65; n₂= 400,  = 0.58                       

• n₁ = 180, = 0.31; n₂= 250,  = 0.25                       

• n₁ = 100, = 0.46; n₂= 120,  = 0.61                       

• Suppose you want to estimate the difference between two population means correct to within 1.8 with a 95% confidence interval.  If prior information suggests that the population variances are approximately equal to Ϭ₁² = Ϭ₂² = 14 and you want to select independent random samples of equal size from the population, how large should the sample sizes, n₁ and n₂be?                             

• Determine each of the following F values:
  • F_.05 where v₁-= 9 and v₂ = 6                        
  • F_.01 where v₁-= 18 and v₂ = 14                   
  • F_.025 where v₁-= 11 and v₂ = 4                    
  • F_.10 where v₁-= 20 and v₂ = 5                      

1. When new instruments are developed to perform chemical analyses of products (food, medicine etc.), they are usually evaluated with respect to two criteria: accuracy and precision.  Accuracy refers to the ability of an instrument to identify correctly the nature and amounts of a product’s components.  Precision refers to the consistency with which the instrument will identify the components of the same material.  Thus, a large variability in the identification of a single batch of a product indicates a lack of precision.  Suppose a pharmaceutical firm is considering two brands of an instrument designed to identify the components of certain drugs.  As part of a comparison of precision, 10 test-tube samples of a well-mixed batch of a drug are selected and then five are analyzed by instrument A and five by instrument B.  The data shown below are the percentages of the primary component of the drug given by the instruments.  Do these data provide evidence of a difference in the precision of the two machines? Use α = 0.10.

Instrument A	Instrument B
43	46
48	49
37	43
52	41
45	48

 

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Market research has indicated that customers are likely to bypass Roma tomatoes that weigh less than 70 grams. A produce company produces Roma tomatoes that average 78.0 grams with a standard deviation of 5.2 grams. 

(a) [2 marks] Assuming that the normal distribution is a reasonable model for the weights of these tomatoes, what proportion of Roma tomatoes are currently undersize (less than 70g)? 

(b) [2 marks] How much must a Roma tomato weigh to be among the heaviest 20%? 

(c) [2 marks] The aim of the current research is to reduce the proportion of undersized tomatoes to no more than 2%. One way of reducing this proportion is to reduce the standard deviation. If the average size of the tomatoes remains 78.0 grams, what must the target standard deviation be to achieve the 2% goal? 

(d) [3 marks] The company claims that the goal of 2% undersized tomatoes is reached. To test this, a random sample of 20 tomatoes is taken. What is the distribution of the number of undersized tomatoes in this sample if the company’s claim is true? Explain your reasoning. 

(e) [3 marks] Suppose there were 3 undersized tomatoes in the random sample of 20. What is the probability of getting at least 3 undersized tomatoes in a random sample of 20 if the company’s claim is true? Do you believe the company’s claim? Why or why not?

 

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