Prehistoric pottery vessels are usually found as sherds (broken pieces) and are carefully reconstructed if enough sherds can be found. Information taken from a study provides data relating x = body diameter in centimeters and y = height in centimeters of prehistoric vessels reconstructed from sherds found at a prehistoric site. The following Minitab printout provides an analysis of the data. Predictor Coef SE Coef T P Constant -0.203 2.429 -0.09 0.929 Diameter 0.7648 0.1771 5.33 0.005 S = 4.28980 R-Sq = 70.3% (a) The standard error $S_e$ of the linear regression model is given in the printout as \"S.\" What is the value of $S_e$? 4.28980 (b) The standard error of the coefficient of the predictor variable is found under \"SE Coef.\" Recall that the standard error for b is $\frac{S_e}{\sqrt{\sum x^2 - \frac{(\sum x)^2}{n}}}$. From the Minitab display, what is the value of the standard error for the slope b?
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4. Consider the "sale.txt" data posted (at onQ, below this file). A company conducted a survey of a new product to develop a marketing strategy. The response Y is the amount (in thousand dollars) that each individual can spend on the new product. The possible explanatory variable x is the yearly income (in hundred dollars). There are 21 participants in the study. (a) Fit a simple linear regression for Y and x. Define this model clearly in mathematical form. Assess the model fit by the 3 types of residual plots introduced in Section 5.2. Do you find any problems with the constant variance assumption? (b) Use the Box-Cox transformation on Y to improve the model. What transformation do you choose? Define a new model in mathematical form based on this transformation. Fit the model, and assess the model performance using suitable residual plot or plots. Is it a good remedy for the problem identified in (a)? (c) Now consider a model without intercept, Yi = ̢̢xi + ̢i where Yi and xi are the amount individual i can spend on the new product, and his/her yearly income. The error terms ̢i are assumed to be i.i.d. N(0, ̢2). Fit this model to the data. Does it fit the data better than the model in (a)? Repeat residual analysis (and plots) for this model as in (a) and comment. (d) Suppose the true model is Yi = ̢xi + ̢i where error terms ̢i are independent from N(0, ̢2xi) distribution. That is, Var(Yi) = Var(̢i) = ̢2xi. Notice this is a model without an intercept, and for heteroscedastic data! A possible remedy for heteroscedasticity is to consider the model Yi* = ̢xi* + ̢i*, where Yi* = Yi/√xi, xi* = √xi and ̢i* = ̢i/√xi. In theory, do ̢i*'s have equal variances now? Fit this model to the data, and assess the model performance through relevant residual plot or plots. Does it work in fixing the non-constant variance problem for the model in (c)? Remark: Part (d) is really applying a weighted least squares method to the heteroscedastic data. We can also apply it directly to the original data through the lm() function with weights. Try this to see if you get exactly the same output as from the model you fit for (d). > fitwt=lm(Y~X-1,weights=1/X) # Original data attached.
Dominador T.
Stat 412 Assignment #4 Do all problems. Due: Wednesday, February 12, 2020 1. A semiconductor manufacturer collected data from a new tool and conducted a hypothesis test with the null hypothesis being a critical dimension mean width equals 100 nm. After the test was carried out, the conclusion was to reject the null hypothesis. Does the test provide strong evidence that the critical dimension mean width equals 100 nm? Explain your answer. 2. The true mean pull-off force of a connector depends upon the cure time. (a) State the null and alternative hypotheses that might be used to demonstrate that the pull-off force is below 25 Newtons. (b) Assume that the test in (a) does not reject the null hypothesis. Does it provide strong evidence that the pull-off force is greater than or equal to 25 newtons? Explain your answer. 3. The heat evolved in calories per gram of a cement mixture is found to follow distribution with standard deviation 2. The cement mixture will be used for a construction project unless evidence suggests that the true mean heat of the mixture is above 100. (a) Suppose mean heat based on a sample of size 25 is found to be 100.5. Formulate the appropriate hypotheses and then conclude if the cement mixture can be used for the project. Use ̑ = 0.05 (b) State the type of error that might have been committed in your conclusion in (a). 4. The tensile strength of a steel alloy intended for use in golf club shafts is known to be normally distributed with standard deviation of 60 psi. Suppose a random sample of 12 specimens has a mean tensile strength of 3450 psi. Is there evidence in data to conclude that the true mean strength differs from 3500 psi and draw your conclusions. Use ̑ = 0.10 5. A bearing used in an automotive application should have an inside diameter that does not differ from 1.5 inches. A random sample of 32 bearings is selected and the average inside diameter of these bearings is 1.4975 inches. You may assume that the bearing diameter is normally distributed and the standard deviation is 0.01 inches. On the basis of the sample information, can you conclude that the true mean diameter of bearings does not meet the required specification? 6. A random sample of 50 batteries is selected and subjected to a life test. The average life of these 50 batteries is 4.05 hours. Assume battery life is normally distributed with standard deviation 0.2 hours. Is there evidence in the data to support the claim that mean battery life exceeds 4 hours? Use ̑ = 0.02
Adi S.
Use technology to find the quadratic regression curve through the given points. (Round all coefficients to four decimal places.) (1, 4), (3, 7), (4, 5), (5, 3) y(x) = Crude oil imports to one country from another for 2009–2013 could be approximated by the following model where t is time in years since the start of 2000. I(t) = -33t^2 + 800t - 3,000 thousand barrels per day (9 ≤ t ≤ 13) According to the model, approximately when were oil imports to the country greatest? (Round your answer to two decimal places.) t = How many barrels per day were imported at that time? (Round your answer to two significant digits.) thousand barrels Revenue The market research department of the Better Baby Buggy Co. predicts that the demand equation for its buggies is given by q = -1.5p + 540 where q is the number of buggies it can sell in a month if the price is $p per buggy. At what price (in dollars) should it sell the buggies to get the largest revenue? p = $ What is the largest monthly revenue (in dollars)? $ Revenue Pack-Em-In has another development in the works. If it builds 40 houses in this development, it will be able to sell them at $260,000 each, but if it builds 70 houses, it will get only $230,000 each. Obtain a linear demand equation. (Let p be the price of a house and q the number of houses.) p(q) = Determine how many houses it should build to get the largest revenue. houses What is the largest possible revenue (in dollars)? $ The following table shows a tech product's sales during the financial years 2005–2009. (t is time in years since 2005.) Year t 0 1 2 3 4 iPod Sales S (millions) 22.8 39.4 51.2 54.8 54.7 (a) Find a quadratic regression model for these data. (Round coefficients to three significant digits.) S(t) = Graph the model together with the data. (b) What does the model predict for the product's sales in 2010, to the nearest million? $ million What does the model predict for the product's sales in 2011, to the nearest million? $ million The product's true sales in 2010 and 2011 were $50.4 million and $42.6 million respectively. Comment on your answers. - The answers we found are within $2 million of the actual sales, so the predictions were reasonably accurate. It is safe to extrapolate the model within two points of the given data. - The answers we found differ from the actual sales by more than $2 million, so the predictions were not reasonably accurate. This shows the danger of extrapolating the model beyond the given data. - The answers we found differ from the actual sales by more than $2 million, so the predictions were reasonably accurate. It is safe to extrapolate the model within two points of the given data. - The answers we found are within $2 million of the actual sales, so the predictions were not reasonably accurate. This shows the danger of extrapolating the model beyond the given data.
Sri K.
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