Please do problems 4 - 7 and place the answers in a word file with the necessary diagrams if asked for (You need information from Problem 3 but you don't have to complete it) You need to use stat crunch so my log in is :username:
[email protected]:W789tyughThe data set and problems are attached below (the data set you have to load into stat crunch)
No referencing required
STAT 202-014 To load data into StatCrunch: Data > Load > From file > On my computer > Choose file > filename > Open > Load data To copy-paste a part of the StatCrunch numerical results or a part of the StatCrunch spreadsheet: highlight the part > Ctr-C > Ctr-V To save a StatCrunch graph in your file: Options > Copy > right click on the image > Copy image > Ctr+Alt+V > Device Independent Bitmap > Ok Z procedures for tests and Cis: Example for a test H0: µ = 120 versus Ha: µ ≠ 75 at ? = 0.05 = 1 – 0.95 and for a CI at c = 0.95 To generate a normal probability plot (QQ plot): enter the sample values in a column and name the column appropriately > Graph > QQ Plot > Select the column > Compute This assignment is composed of three parts. Part 1 (2 problems) Part 2 (5 problems) Part 3 (3 problems) Please SHOW YOYR WORK everywhere it is requested. Answers given without supporting calculations will have no grade value. Part 1 Problem 1 (worth 5 points) The demand for a certain weekly magazine at a newsstand is a discrete random variable, X, with an expected value of 3 magazines sold per week. The demand never exceeds 6 magazines per week. Furthermore, the distribution of variable X is symmetric about the value of 3. (a) The table below is intended to present the distribution of variable X. Complete the table. Justify your values. x (value of X) 0 1 2 3 4 5 6 Probability of x 0.05 0.10 0.20 (b) The magazines cost $4.00 per copy for the owner of the newsstand and are sold for $6.00 per copy to the customers. At the beginning of each week, the owner of the newsstand buys 6 magazines to sell during the week. In dollars, what is the expected amount of money the owner of the newsstand will take in from the sales of the magazines per week? Show your work. [Hint: maybe, first, you want to present a table similar to the one above] (c) Explain briefly why it would not be wise for the owner of the newsstand to buy 6 magazines at the beginning of each week. Problem 2 (worth 8 points) The BioPharm company has developed a stress test for college-aged students. Scores on the stress test for such students are approximately normally distributed with a mean of 55 points and a standard deviation of 2 points. Scores of 58 points or higher indicate a high level of stress and are of concern to doctors. A random sample of college-aged students will be selected and each will be given this stress test. Answer the following questions based on the information given. (a) What is the probability that the score for the first randomly selected student will be at least 58 points? Show your work. (b) Given that the first randomly selected student has a score of at least 58 points, what is the probability that the next randomly selected student scores below 58 points? Show your work. (c) For each randomly selected sample of six college-aged students variable Y yields the number of students in the sample with a high level of stress (a score of at least 58 points.) What kind of a variable is Y? Clearly indicate your choice. (A) Y is a normal variable with a mean of 55 and a standard deviation of 2. (B) Y is a normal variable with a mean of 55 and standard deviation of 12 (6 students times 2). (C) Y is an binomial variable with n = 6 and p = probability that the score is at least 58 points. (D) Y is a binomial variable with n = 6 and p = 0.5. (d) Suppose the stress test is given to six randomly selected college-aged students. What is the probability that exactly two of the six students will have a high level of stress (a score of at least 58 points)? Show your work. Part 2 Problem 3 (worth 5 points) A sub-critical annealing is a process which makes a steel material tougher. An engineer selects a random sample of 25 ductile iron pieces which were sub-critically annealed and measures their Brinell hardness numbers. The resulting data is posted on Blackboard, in Data files, as BrinellHardnessNumber.xlxs. (Use exclusively this file, not any other file from a different source). The engineer hypothesizes that the mean Brinell hardness number for all sub-critically annealed ductile iron pieces is greater than 170 compared with the mean Brinell hardness number of 170 for unprocessed pieces. To investigate the engineer’s suggestion, you want to conduct an appropriate statistical test for the mean Brinell hardness number for all sub-critically annealed ductile iron pieces. You know that certain assumptions need to be met for conducting your test. One of these assumptions is that variable X, the Brinell hardness number of a piece, must be normal or near normal when the sample is small as in this case (n = 25 < 30). (a)* to check whether the sample data, although discrete, can be viewed as coming from a normal or near normal distribution, using statcrunch, construct a normal probability plot for the data. present the original statcruch plot. (b) refer to the plot presented in (a)? do the sample values seem to be coming, approximately, from a normal distribution? justify your answer. [note: the horizontal clusters of points are okay. they are due to the fact the x is actually a discrete variable taking integers. however, if the points approximately follow the straight line seen in the diagram then x can be viewed as normal and can be approximated by a normal distribution, in our course without a continuity correction.] problem 4 (worth 15 points!!!) refer to problem 3. view variable x as normal. to investigate the engineer’s suggestion conduct an appropriate statistical test for the mean brinell hardness number, ?, for the population of all sub-critically annealed ductile iron pieces. assume that the population standard deviation is ? = 0.3. conduct, by hand, the test at a significance level of 0.05. show your work organized in 6 steps. use the appropriate color to mark the p-value area. [hint: for the sample mean use your calculator or statcrunch. you do not need to show how you calculated the sample mean.] problem 5 (worth 5 points) refer to problems 3 and 4. (a) under the assumptions of problems 3 and 4 calculate, by hand, a 95% confidence interval for the mean brinell hardness of all ductile iron pieces sub-critically annealed. show your work. include an appropriate diagram. (b) interpret the interval calculated in (a). what does this interval tell you about the mean hardness number for ductile iron pieces sub-critically annealed? problem 6* (worth 4 points) refer to the test conducted in problem 4. conduct the same test using statchrunch. present the relevant statcrunch table and point to the tests statistic value and the p-value. is the statcrunch result different from or the same as your result in problem 4, and why? [hint: see page 2 of this assignment for statcrunch instructions.] problem 7* (worth 3 points) refer to the confidence interval calculated in problem 5. calculate the same confidence interval using statcrunch. present the relevant statcrunch table and point to the interval. is the statcrunch result different from or the same as your result in problem 5, and why? part 3 problem 8 (worth extra 5 points) under the assumptions of problems 3 and 4 calculate, by hand, a 65% confidence interval for the mean brinell hardness of all ductile iron pieces sub critically annealed. show your work. include an appropriate diagram. interpret the interval. are you satisfied with this interval? justify your answer. problem 9 (worth extra 7 points) you would like to conduct a statistical test for a population parameter at a significance level of ? = 0.1. under the assumption that the null hypothesis is true the test statistic for the test has the distribution shown below. at this moment you do not know how to handle such cases. don’t worry. this problem is not about conducting a test. this problem is about establishing the rejection region for the test. (a) copy-paste the distribution diagram from the file0 teststatisticdistribution.docx (posted on blackboard in data files) to your page. (b) calculate the area under the distribution. show your work. (c) using a red pencil mark an area of ?/2 in the left tail, and an area of ?/2 in the right tail. [hint: the red area in the left tail will have a 30).="" (a)*="" to="" check="" whether="" the="" sample="" data,="" although="" discrete,="" can="" be="" viewed="" as="" coming="" from="" a="" normal="" or="" near="" normal="" distribution,="" using="" statcrunch,="" construct="" a="" normal="" probability="" plot="" for="" the="" data.="" present="" the="" original="" statcruch="" plot.="" (b)="" refer="" to="" the="" plot="" presented="" in="" (a)?="" do="" the="" sample="" values="" seem="" to="" be="" coming,="" approximately,="" from="" a="" normal="" distribution?="" justify="" your="" answer.="" [note:="" the="" horizontal="" clusters="" of="" points="" are="" okay.="" they="" are="" due="" to="" the="" fact="" the="" x="" is="" actually="" a="" discrete="" variable="" taking="" integers.="" however,="" if="" the="" points="" approximately="" follow="" the="" straight="" line="" seen="" in="" the="" diagram="" then="" x="" can="" be="" viewed="" as="" normal="" and="" can="" be="" approximated="" by="" a="" normal="" distribution,="" in="" our="" course="" without="" a="" continuity="" correction.]="" problem="" 4="" (worth="" 15="" points!!!)="" refer="" to="" problem="" 3.="" view="" variable="" x="" as="" normal.="" to="" investigate="" the="" engineer’s="" suggestion="" conduct="" an="" appropriate="" statistical="" test="" for="" the="" mean="" brinell="" hardness="" number,="" ,="" for="" the="" population="" of="" all="" sub-critically="" annealed="" ductile="" iron="" pieces.="" assume="" that="" the="" population="" standard="" deviation="" is="" =="" 0.3.="" conduct,="" by="" hand,="" the="" test="" at="" a="" significance="" level="" of="" 0.05.="" show="" your="" work="" organized="" in="" 6="" steps.="" use="" the="" appropriate="" color="" to="" mark="" the="" p-value="" area.="" [hint:="" for="" the="" sample="" mean="" use="" your="" calculator="" or="" statcrunch.="" you="" do="" not="" need="" to="" show="" how="" you="" calculated="" the="" sample="" mean.]="" problem="" 5="" (worth="" 5="" points)="" refer="" to="" problems="" 3="" and="" 4.="" (a)="" under="" the="" assumptions="" of="" problems="" 3="" and="" 4="" calculate,="" by="" hand,="" a="" 95%="" confidence="" interval="" for="" the="" mean="" brinell="" hardness="" of="" all="" ductile="" iron="" pieces="" sub-critically="" annealed.="" show="" your="" work.="" include="" an="" appropriate="" diagram.="" (b)="" interpret="" the="" interval="" calculated="" in="" (a).="" what="" does="" this="" interval="" tell="" you="" about="" the="" mean="" hardness="" number="" for="" ductile="" iron="" pieces="" sub-critically="" annealed?="" problem="" 6*="" (worth="" 4="" points)="" refer="" to="" the="" test="" conducted="" in="" problem="" 4.="" conduct="" the="" same="" test="" using="" statchrunch.="" present="" the="" relevant="" statcrunch="" table="" and="" point="" to="" the="" tests="" statistic="" value="" and="" the="" p-value.="" is="" the="" statcrunch="" result="" different="" from="" or="" the="" same="" as="" your="" result="" in="" problem="" 4,="" and="" why?="" [hint:="" see="" page="" 2="" of="" this="" assignment="" for="" statcrunch="" instructions.]="" problem="" 7*="" (worth="" 3="" points)="" refer="" to="" the="" confidence="" interval="" calculated="" in="" problem="" 5.="" calculate="" the="" same="" confidence="" interval="" using="" statcrunch.="" present="" the="" relevant="" statcrunch="" table="" and="" point="" to="" the="" interval.="" is="" the="" statcrunch="" result="" different="" from="" or="" the="" same="" as="" your="" result="" in="" problem="" 5,="" and="" why?="" part="" 3="" problem="" 8="" (worth="" extra="" 5="" points)="" under="" the="" assumptions="" of="" problems="" 3="" and="" 4="" calculate,="" by="" hand,="" a="" 65%="" confidence="" interval="" for="" the="" mean="" brinell="" hardness="" of="" all="" ductile="" iron="" pieces="" sub="" critically="" annealed.="" show="" your="" work.="" include="" an="" appropriate="" diagram.="" interpret="" the="" interval.="" are="" you="" satisfied="" with="" this="" interval?="" justify="" your="" answer.="" problem="" 9="" (worth="" extra="" 7="" points)="" you="" would="" like="" to="" conduct="" a="" statistical="" test="" for="" a="" population="" parameter="" at="" a="" significance="" level="" of="" =="" 0.1.="" under="" the="" assumption="" that="" the="" null="" hypothesis="" is="" true="" the="" test="" statistic="" for="" the="" test="" has="" the="" distribution="" shown="" below.="" at="" this="" moment="" you="" do="" not="" know="" how="" to="" handle="" such="" cases.="" don’t="" worry.="" this="" problem="" is="" not="" about="" conducting="" a="" test.="" this="" problem="" is="" about="" establishing="" the="" rejection="" region="" for="" the="" test.="" (a)="" copy-paste="" the="" distribution="" diagram="" from="" the="" file0="" teststatisticdistribution.docx="" (posted="" on="" blackboard="" in="" data="" files)="" to="" your="" page.="" (b)="" calculate="" the="" area="" under="" the="" distribution.="" show="" your="" work.="" (c)="" using="" a="" red="" pencil="" mark="" an="" area="" of="" 2="" in="" the="" left="" tail,="" and="" an="" area="" of="" 2="" in="" the="" right="" tail.="" [hint:="" the="" red="" area="" in="" the="" left="" tail="" will="" have="">