STAT 2606: Business Statistics I – Assignment 4

QUESTION

Carleton UniversitySchool of Mathematics and StatisticsSTAT 2606: Business Statistics I – Assignment 4Section C due Tuesday, November 17 at 2:25pm.Section D due Wednesday, November 18 at 2:25pm.INSTRUCTIONS:I)Assignments are to be submitted in-class on the due date and prior to beginning of the lecture. Nolate assignments will be accepted without sufficient advanced noticed and a legitimate,II)documented reason.You must show all of your work. No credit will be given for answers without justification. You mustIII)explicitly define any random variable you use in a solution. No credit will be given for illegible work.Do not use MINITAB for any part of a question unless it specifically says to do so. For questionsIV)V)that require MINITAB, you must include all relevant output with your assignment.The lab for this assignment will take place during the week of November 9.This assignment is intended to represent your individual knowledge. It is not a group assignment.Part AMINITAB QUESTIONSQuestion 1. [Central limit theorem (CLT)] First, enable commands in the SESSION window asfollows:CLICK Editor, then CLICK Enable CommandsYou should now see an MTB > prompt in the session window. If so, you are ready to begin typing,into the SESSION window, the commands in the boxes below.Generate and store, in columns c3-c902, a total of 100 row-wise samples, each of size n = 900, froma Poisson distribution with parameter λ = 9 as follows (Note that there is a semi-colon at the end of thefirst line, and a period at the end of the second):random 100 c3-c902;poisson 9.1Note This may take a few moments as you are generating 900 × 100 = 90, 000 values.Create and store in column c1 the 100 values of X based on the 100 samples of same size n = 900 asfollows:rmean c3-c902 c1(a) Print (and include in your assignment) the boxplot of c3. According to the position of the median,what can you conclude about the shape of this data set?boxplot c3(b) Use the describe command to find the sample mean and standard deviation of c3 by typing thefollowing command in the SESSION window:desc c3(c) Print (and include in your assignment) the boxplot for the data in column c1. What can you concludeabout the shape of data in c1?boxplot c1(d) Use the describe command to find the sample mean and standard deviation of c1 by typing thefollowing command in the SESSION window:desc c1Are they close to 9 and 3/30? Why?Before starting question 2, you should open a new MINITAB Worksheet(Select File → New→ Minitab Worksheet) or erase the data from question 1 by typing the following commandin the session window:erase c1-c1000Question 2. [Confidence interval (CI) for a mean:] Similar to QUESTION 1, you will betyping the commands in the boxes below into the SESSION window in MINITAB.We want to build 100 confidence intervals (CIs) with confidence level (1 − α)100% = 95% for the meanµ of a Normal distribution via the following steps:Step 1. Generate and store in columns c6-c30 100 samples of size 25 each from Normal with parameterµ = 2, σ = 5 as follows:Page 2random 100 c6-c30;normal 2 5.Step 2. Use columns c4 and c5 to store respectively the means and the standard deviations of the 100samples you generated in step 1, as follows:rmean c6-c30 c4rstd c6-c30 c5Step 3. Store the lower bound and the upper bound of your 95% CIs in c2 and c3 respectively bytyping successively:let c2=c4-1.96*c5/5let c3=c4+1.96*c5/5Then create a column c1 containing 1 or 0 according to whether the corresponding interval [c2 , c3]covers µ or not, by typing:let c1=(c2 <= 2 and c3 >= 2)Finally sum up the entries of column c1 to find how many CIs cover the value µ = 4 by typing:tally c1(a) What is the percentage of confidence intervals that contain the true value µ = 2?(b) How do you compare this percentage to the confidence level 95%?Page 3Part BWritten QuestionsQuestion 1. Suppose a sample of 120 items is drawn from a population of manufactured products andthe number of defective items is recorded. Prior experience has shown that the proportion of defectives is0.05.(a) Describe the sampling distribution of p, the proportion of defectives.(b) What is the probability that the sample proportion is less than 0.10?Question 2. Statistics suggest that software developers with two years of experience in Ontario earn anaverage of 70, 000 per year, with a standard deviation of 5, 000. To verify this salary level, a randomsample of 100 software developers with two years of experience was selected from a personnel database forall software developers in Ontario.¯(a) Describe the sampling distribution of the sample mean X of the average salary of these 100 SoftwareDeveloper.(b) Calculate the probability that X is greater than or equal to 71,200.(c) If the random sample actually produced a sample mean of 71,200, would you consider this ratherunusual? What conclusion might you draw then?Question 3. The college president asks the statistics teacher to estimate the average age of the studentsat their college. How large a sample is necessary? The statistics teacher would like to be 99% confidentthat the estimate should be accurate within 0.5 year. From a previous study, the standard deviation ofthe ages is known to be 2 years.Question 4. Suppose that interest lies in estimating the proportion of drivers who speed on the highway.What sample size is needed to estimate this proportion to within 5% with 90% confidence?Question 5. The economy suffered significantly during 2008. There were a large number of foreclosuresof family homes. In one large community, realtors randomly sampled 36 bids from potential buyers todetermine the average loss in home value. The sample showed the average loss was $11,560. Suppose it isknown that σ = $1500.(a) Find a 80% confidence interval for the mean loss in value per home.(b) Interpret this interval in (a) and explain what 80% confidence means.(c) Is the estimate in part (a) valid? Explain.Question 6. In a survey of 1,000 people, 420 are opposed to the tax increase.(a) Construct a 95 percent confidence interval for the proportion of those people opposed to the taxincrease.(b) Interpret the CI in terms of the question.(c) Is the estimate in part (a) valid? Explain.Question 7. A sample of 8 items has an average fat content of 18.6 grams and a sample standard deviationof 2.4 grams. Assuming a normal distribution.(a) Construct a 99 percent confidence interval for the population mean.Page 4(b) Interpret the CI in terms of the question.(c) Is the estimate in part (a) valid? Explain.Page 5

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