Exploring Sampling Distributions and the Central Limit Theorem Using Minitab Lab 2

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Page1of7Lab 2Exploring Sampling Distributions and the Central Limit Theorem Using MinitabLab 2Objectives:Learn to use Minitab to compute probabilities and percentile of distributions.Use simulation to understand central limit theorem andsampling distributions1.Using Minitab, answer the following. Write your answers in your document that you will submit.Your answer must be complete. Remember that Minitab only helps you to calculate your answer.You need to provide all the details such as proper definitions, proper set-up of the problem andexplanation of the process. You will not get much credit without all this information.The time required for Speedy Lube to complete an oil change service on an automobile follows anormal distribution, with a mean of 17 minutes and a standard deviation of 2.5 min.Question 1a:Speedy Lube guarantees customers that the service will take no longer than 20 minutes. If itdoes take longer, the customer will receive the service for half-price. What percent ofcustomers receive the service for half-price?Solution:Let X be the random variable then the probability will beP(X>20)=P(𝑋−1଻2.5>20−1଻2.5)=𝑃(𝑍>1.2)=0.1151where Z=𝑋−1଻2.5Hence 11.51% of customer receives the service for half price.Question 1b:If Speedy Lube does not want to give the discount to more than 3% of its customers, howlong should it make the guaranteed time limit?Solution:P(Z≥𝑍0.03)≤0.03P(𝑋−1଻2.5≥1.181)≤0.03P(X≥19.9525)≤0.03Hence the granted limit is19.9525.Question 1c:What fraction of autos take between 11 min. and 16 min.?Solution:P(11≤𝑋≤16)=𝑃(11−1଻2.5≤𝑋−1଻2.5≤16−1଻2.5)=𝑃(−2.4≤𝑍≤−0.4)=0.33642.Next we see the idea of using simulation to generate a sample from a population.The populationdistribution is specified and we draw a sample from this population by generating randomnumbers from the population distribution.We will learn about the behavior of the statisticscomputed for several samples from the population.Thesampling distributionof a statistic is the distribution of values taken by the statistic in allpossible samples of the same size from the same population. Recall that astatisticis a measure of asample, whereas aparameteris a measure of a population.

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Page2of7Lab 2Suppose the number of toys a child plays with in a child care center on a given day is a random variable.Suppose the probability distribution of X is given byx012345P(X=x)0.030.060.10.130.470.21i.ChooseCalc--> Setbaseand enter the l yourstudent identity number(notyour socialsecurity number). Click “OK”.ii.Enter the values 0,1,2,3,4,5 in column c1.Name this columnx.iii.Enter the values 0.03, 0.06, 0.1, 0.13, 0.47, 0.21 in column c2.Name this columnP(X=x).So, the pmf of X is given in columns C1 and C2. Let us consider this data , the number of toys a childplays with, as the population. So, the population contains values 0,1,2,3,4,5 in the relative frequency givenin C2. Columns C1 and C2 provide the population distribution.iv.Graph this population distribution usingGraph--> Scatterplot,selecting theSimplegraphoption; choose the column with values of the random variable as thex variableand the columnwith the probabilities as they variable.Click theDataviewbutton and in theDataviewmenu,selectonlyproject lines.Copy and paste(special) the graph in your document.The graph is as follows:Question 2a.: Comment on the shape of the(population) distribution that is represented inyour graph.5432100.50.40.30.20.10.0xP(X=x)Scatterplot of P(X=x) vs x

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