1. Normal Distribution: A company supplies pins in bulk to a customer. The company uses an automatic lathe to produce the pins. Due to many causes -vibration, temperature, wear and tear, and the like- the lengths of the pins made by the machine are normally distributed with a mean of 1.012 inches and a standard deviation of 0.018inch. The customer will buy only those pins with length of 1.00inch but will accept up to 0.02inch deviation on either side. This 0.02 is known as the tolerance.
(a) What percentage of the pins will be acceptable to the consumer?
In order to improve percentage accepted, the production manager and the engineers discuss adjusting the population mean and the standard deviation of the length of the pins.
(b) If the lather can be adjusted to have the mean of the lengths equal to any desired value, what should it be adjusted to? Why?
(c) Suppose the mean cannot be adjusted but the standard deviation can be reduced. What maximum value of the standard deviation would make 90% of the parts acceptable to the consumer? (Assume the mean to be 1.012)
(d) Repeat question (c), with 95% and 99% of the pins acceptable.
(e) In practice, which one do you think is easier to adjust; the mean or the standard deviation?
Why?
The manager then considers the costs involved. The cost of resetting the machine to adjust the population mean involves the engineers' time and the cost of production time lost. The cost of reducing the population standard deviation involves, in addition to these costs, the cost of overhauling the machine and reengineering the process.
(f) Assume it costs $150 x2 to decrease the standard deviation by (x/1000) inch. Find the cost of reducing the standard deviation to the values found in questions (c) and (d).
(g) Now assume that the mean has been adjusted to the desired value found in question (b) at a cost of $80. Calculate the reduction in standard deviation necessary to have 90%, 95% and 99% of the parts acceptable. Calculate the respective costs, as in question (f).
(h) Based on your answers to questions (f) and (g), what are your recommended mean and standard deviation?
2. Confidence Intervals: A company wants to conduct a telephone survey of randomly selected voters to estimate the proportion of voters who favor a particular candidate in a presidential election, to within 2% error with 95% confidence1. It is guessed that the proportion is 53%.
(a) What is the required minimum sample size?
(b) The project manager assigned to the survey is not sure about the actual proportion or about the 2% error limit. Construct a table for the minimum sample size required with margin of error ranging from 1% to 3% and actual proportion ranging from 40% to 60%.
(c) Inspect the table produced in question (b) above. Comment on the relative sensitivity of the minimum sample size to the actual proportion and to the desired margin of error.
(d) At what value of the actual proportion is the required sample size the maximum?
(e) The cost of polling includes a fixed cost of $425 and a variable cost of $1.20 per person sampled, thus the cost of sampling n voters is $(425+1.20n). Tabulate the cost for the range of values in question (b) above.
(f) A competitor of the company that had announced results to within ±3% with 95% confidence has started to announce results to within ±2% with 95% confidence. The project manager wants to go one better by improving the company's estimate to be within ±1% with 95% confidence. What would you tell the manager?
1 The 2% error is in absolute value, i.e., the 95% C.I. for the estimated proportion should be proportion±2%.
3. Non-linear Transformation: Influential wine critics such as Robert Parker publish their personal ratings of wines and many consumers pay close attention. Do these ratings affect the price? The data in wine.xlsx are a sample of ratings and prices found online at the website of an internet wine merchant.
(a) Does the scatterplot of the price of wine on the rating suggest a linear or nonlinear relationship?
(b) Fit a linear regression equation to the data, regressing price on the rating. Does this fitted model make substantive sense?
(c) Create a scatterplot for the natural logarithm of the price on the rating. Does the relationship seem more suited to regression?
(d) Fit a regression of the natural logarithm of price on the rating. Does this model provide a better description of the pattern in the data?
(e) Compare the fit of the two models to the data.
4. Multiple Regression: The data in apple.xlsx tracks monthly performance of stock in Apple Computer since its reception in 1980. The data include 300 monthly returns on Apple Computer, IBM stock returns as well as returns on the entire stock market. Formulate the model with Apple Return as the response and Market Return and IBM Return as explanatory variables. (a) Examine scatterplots of the response versus the two explanatory variables as well as the scatterplot between the responses. Do you notice any unusual features in the data? Do the relevant plots appear straight enough for multiple regression?
(b) Fit the indicated multiple regression and show a summary of the estimated features of the model.
(c) The regression of Apple returns on market returns estimates β for this stock to be about 1.5.
Does the multiple regression suggest a different slope for the market?
(d) Give a confidence interval for the coefficient of IBM returns and carefully interpret this estimate.
(e) Does the inclusion of IBM returns improve the fit of the model with just market returns by a statistically significant amount? Does this imply that we've found an improved trading scheme?
Attachment:- Assignment.rar