# Lab 2 | Complete Solution

Lab 2

Objectives:

Learn to use Minitab to compute probabilities and percentile of distributions.

Use simulation to understand central limit theorem and sampling 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 and explanation 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 a normal 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 it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half-price?

Question 1b: If Speedy Lube does not want to give the discount to more than 3% of its customers, how long should it make the guaranteed time limit?

Question 1c: What fraction of autos take between 11 min. and 16 min.?

Next we see the idea of using simulation to generate a sample from a population. The population distribution is specified and we draw a sample from this population by generating random numbers from the population distribution. We will learn about the behavior of the statistics computed for several samples from the population.

The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Recall that a statistic is a measure of a sample, whereas a parameter is a measure of a population.

Suppose 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 by

x

0

1

2

3

4

5

P(X=x)

0.03

0.06

0.1

0.13

0.47

0.21

i. Choose Calc -- Setbase and enter the l your student identity number (not your social security number). Click “OK”.

ii. Enter the values 0,1,2,3,4,5 in column c1. Name this column x .

iii. Enter the values 0.03, 0.06, 0.1, 0.13, 0.47, 0.21 in column c2. Name this column P(X=x).

So, the pmf of X is given in columns C1 and C2. Let us consider this data , the number of toys a child plays with, as the population. So, the population contains values 0,1,2,3,4,5 in the relative frequency given in C2. Columns C1 and C2 provide the population distribution.

iv. Graph this population distribution using Graph-- Scatterplot, selecting the Simple graph option; choose the column with values of the random variable as the x variable and the column with the probabilities as the y variable. Click the Dataview button and in the Dataview menu, select only project lines. Copy and paste(special) the graph in your document.

Question 2a.: Comment on the shape of the( population) distribution that is represented in your graph.

v. Compute the mean and variance of the population. Copy these values in your document clearly identifying what they measure and describing them. Remember the formulas for computing the mean of a discrete distribution: µ = E(X) = S xP(X=x) and

σ 2= S x2P(X=x) - µ2.

Question 2b: Write the mean and variance of the population in your document using appropriate notations, units, and with a short description of what they stand for.

vi. Name columns C3, C4, and C5, as smeans_25, smeans_64, smeans_100, .

vii. You need to generate 1000 rows (samples); each of size 25. Use the Calc Random Data Discrete command to Generate 1000 samples of 25 values from our population, storing the results in columns c7-C31. Remember that the values of the data are in column c1 and the probabilities are in c2. You have just simulated observing 25 children and noting how many toys each child plays with and you have repeated this on 1000 occasions! Note that data in columns c7 to c31 in each row is one sample of size 25.

viii.Use the Calc Row Statistics command, with the statistic as the Mean, C7-C31 (your sample) as the input variables and store the results in the column C3 titled smeans_25. The entries in C3 will be the sample means for each of the 1000 samples of size 25.

ix. Create a Graphical summary using Stat-- Basic Statistics -- Graphical Summary of this new column smeans_25. This is the graphical summary for the means of samples of size 25. In the resulting histogram, double click on the x-axis. You should get an Edit Scale dialog box. Set the minimum to 0 and maximum to 5. Double click on the title and retitle the display as “Summary for .....” filling in the blanks to appropriately reflect what it is displaying. The distribution that you see is an example of sampling distribution since it is a distribution of the sample mean (Statistic).

Paste the graph to your document; comment on the shape and features of the distribution of means of samples of size 25.

Question 2c: From the Graphical summary report, find the mean and standard deviation of sample means based on samples of size 25. Write them as such in your report.

Activity 5: Redo parts (vii), (viii) (ix) and Question 2c for sample of size 64. You will still be taking 1000 samples of this size. Each sample will go this time into columns c7-c70 (there are 64 columns!) in part (vii). The sample means will go into column c4 which you had already named as smeans_64, for part (viii). Don’t forget to set the scale minimum and maximum on the x-axis of the graphical display and to copy this graph to your Word document. Write a sentence describing the shape and features of this sampling distribution indicating its mean and standard deviation (smeans_64).

Activity 6: Redo parts (vii), (viii) (ix) and Question 2c for sample of size 100. You will still be taking 1000 samples of this size. Each sample will go into columns c7-c106, for part (vii). The sample means will go into column c5 which you had already named as smeans_100, for part (viii). Don’t forget to set the scale minimum and maximum of the graphical display and to copy this graph to your Word document. Write a sentence describing the shape of this sampling distribution indicating its mean and standard deviation (smeans_100).

Question 2d: Prepare a table similar to the following one in your document and fill-in the details:

Experiment #

Sample size

Mean of sample means

S.D. of sample means

1

2

3

Question 2e:

(i) What were the mean and s.d. of the population? Remember, this was computed earlier. Write these in a sentence form in your document. (don't forget what you are measuring).

(ii) What seems to be happening to the means of the sample means as the sample size gets larger?

Do you detect any pattern of change for the standard deviations of the sample means as the sample size gets larger? Can you identify this pattern with the sample size? Can you describe this behavior in general, for a sample of size n?

(iii) What seems to be happening to the sampling distributions as the sample size gets larger?

What type of population is our sample coming from? This behavior of the sampling distribution is precisely what the central limit theorem guarantees.

.

Question 2f: Comment on the normality of the data on the sample means based on samples of size 25, 64 and 100 by using the normality test.

· Save your word document as Lab02.doc. Print and submit the report before you leave. Make sure that your report is neat and compact. Format it nicely so that there are no empty spaces and the report is no longer than two or three pages. Please don't reprint the data. Your report should only contain answers to the above questions supported by appropriate output from Minitab. Submit your report at the beginning of the class. If you have more than one sheet, make sure that they are stapled.

Objectives:

Learn to use Minitab to compute probabilities and percentile of distributions.

Use simulation to understand central limit theorem and sampling 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 and explanation 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 a normal 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 it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half-price?

Question 1b: If Speedy Lube does not want to give the discount to more than 3% of its customers, how long should it make the guaranteed time limit?

Question 1c: What fraction of autos take between 11 min. and 16 min.?

Next we see the idea of using simulation to generate a sample from a population. The population distribution is specified and we draw a sample from this population by generating random numbers from the population distribution. We will learn about the behavior of the statistics computed for several samples from the population.

The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Recall that a statistic is a measure of a sample, whereas a parameter is a measure of a population.

Suppose 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 by

x

0

1

2

3

4

5

P(X=x)

0.03

0.06

0.1

0.13

0.47

0.21

i. Choose Calc -- Setbase and enter the l your student identity number (not your social security number). Click “OK”.

ii. Enter the values 0,1,2,3,4,5 in column c1. Name this column x .

iii. Enter the values 0.03, 0.06, 0.1, 0.13, 0.47, 0.21 in column c2. Name this column P(X=x).

So, the pmf of X is given in columns C1 and C2. Let us consider this data , the number of toys a child plays with, as the population. So, the population contains values 0,1,2,3,4,5 in the relative frequency given in C2. Columns C1 and C2 provide the population distribution.

iv. Graph this population distribution using Graph-- Scatterplot, selecting the Simple graph option; choose the column with values of the random variable as the x variable and the column with the probabilities as the y variable. Click the Dataview button and in the Dataview menu, select only project lines. Copy and paste(special) the graph in your document.

Question 2a.: Comment on the shape of the( population) distribution that is represented in your graph.

v. Compute the mean and variance of the population. Copy these values in your document clearly identifying what they measure and describing them. Remember the formulas for computing the mean of a discrete distribution: µ = E(X) = S xP(X=x) and

σ 2= S x2P(X=x) - µ2.

Question 2b: Write the mean and variance of the population in your document using appropriate notations, units, and with a short description of what they stand for.

vi. Name columns C3, C4, and C5, as smeans_25, smeans_64, smeans_100, .

vii. You need to generate 1000 rows (samples); each of size 25. Use the Calc Random Data Discrete command to Generate 1000 samples of 25 values from our population, storing the results in columns c7-C31. Remember that the values of the data are in column c1 and the probabilities are in c2. You have just simulated observing 25 children and noting how many toys each child plays with and you have repeated this on 1000 occasions! Note that data in columns c7 to c31 in each row is one sample of size 25.

viii.Use the Calc Row Statistics command, with the statistic as the Mean, C7-C31 (your sample) as the input variables and store the results in the column C3 titled smeans_25. The entries in C3 will be the sample means for each of the 1000 samples of size 25.

ix. Create a Graphical summary using Stat-- Basic Statistics -- Graphical Summary of this new column smeans_25. This is the graphical summary for the means of samples of size 25. In the resulting histogram, double click on the x-axis. You should get an Edit Scale dialog box. Set the minimum to 0 and maximum to 5. Double click on the title and retitle the display as “Summary for .....” filling in the blanks to appropriately reflect what it is displaying. The distribution that you see is an example of sampling distribution since it is a distribution of the sample mean (Statistic).

Paste the graph to your document; comment on the shape and features of the distribution of means of samples of size 25.

Question 2c: From the Graphical summary report, find the mean and standard deviation of sample means based on samples of size 25. Write them as such in your report.

Activity 5: Redo parts (vii), (viii) (ix) and Question 2c for sample of size 64. You will still be taking 1000 samples of this size. Each sample will go this time into columns c7-c70 (there are 64 columns!) in part (vii). The sample means will go into column c4 which you had already named as smeans_64, for part (viii). Don’t forget to set the scale minimum and maximum on the x-axis of the graphical display and to copy this graph to your Word document. Write a sentence describing the shape and features of this sampling distribution indicating its mean and standard deviation (smeans_64).

Activity 6: Redo parts (vii), (viii) (ix) and Question 2c for sample of size 100. You will still be taking 1000 samples of this size. Each sample will go into columns c7-c106, for part (vii). The sample means will go into column c5 which you had already named as smeans_100, for part (viii). Don’t forget to set the scale minimum and maximum of the graphical display and to copy this graph to your Word document. Write a sentence describing the shape of this sampling distribution indicating its mean and standard deviation (smeans_100).

Question 2d: Prepare a table similar to the following one in your document and fill-in the details:

Experiment #

Sample size

Mean of sample means

S.D. of sample means

1

2

3

Question 2e:

(i) What were the mean and s.d. of the population? Remember, this was computed earlier. Write these in a sentence form in your document. (don't forget what you are measuring).

(ii) What seems to be happening to the means of the sample means as the sample size gets larger?

Do you detect any pattern of change for the standard deviations of the sample means as the sample size gets larger? Can you identify this pattern with the sample size? Can you describe this behavior in general, for a sample of size n?

(iii) What seems to be happening to the sampling distributions as the sample size gets larger?

What type of population is our sample coming from? This behavior of the sampling distribution is precisely what the central limit theorem guarantees.

.

Question 2f: Comment on the normality of the data on the sample means based on samples of size 25, 64 and 100 by using the normality test.

· Save your word document as Lab02.doc. Print and submit the report before you leave. Make sure that your report is neat and compact. Format it nicely so that there are no empty spaces and the report is no longer than two or three pages. Please don't reprint the data. Your report should only contain answers to the above questions supported by appropriate output from Minitab. Submit your report at the beginning of the class. If you have more than one sheet, make sure that they are stapled.

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