AluminiCorp is a major producer of aluminum cans that produces 40 billion aluminum cans every year. You work as a quality control officer for AluminiCorp, and are responsible for ensuring that the aluminum cans produced meet certain specifications. Each can is supposed to consist of precisely 15 grams of aluminum, and the market price for aluminum is $0.95 per pound (important note: there are 454 grams in one pound). This file contains data on samples taken from multiple plants under your purview. Use the data to answer the following questions. Problems 3.1 and 3.2 come from week 7 material, and 3.3 relates to week 8 material. Note that part b (and ONLY part b) of question 3.1 is extra credit. Should you encounter any difficulties with these problems, the optional problems below are very similar to the questions in this problem set, and the answers to the optional questions can be found in the back of the textbook. You can also request that the tutor work extensively with you on the optional problems.
Problem 3.1
"Column A contains aluminum content data from a sample of 100 cans taken from your Akron plant. The Akron plant produces 2 billion cans every year.
a) Construct a 95% confidence interval for the population mean aluminum content.
b) EXTRA CREDIT-Construct a 95% confidence interval for the total expenditure on aluminum at the Akron plant. How much money could AluminiCorp save if the Akron plant actually produced cans that were at the target aluminum content?
c) Based on your answer in part (a) (and part (b) if you did that), explain (in plain English) what these results mean.
Hint: make sure you keep your units of measurement consistent (grams v. pounds)!
Excel Tips: The Excel functions =TDIST and =TINV are similar to the functions =NORMSDIST and =NORMSINV and can be used to calculate T critical values. See the Excel helpfile for instructions on the syntax for this function.
General tip for this problem. In part (a), you calculated a confidence interval for aluminum content. The units of your mean and standard deviation are grams per can. For part (b), you need to first convert your sample mean and standard deviation from part (a) from grams per can into dollars per can. "
Problem 3.2
We do not have access to your data.
In Problem 3.1, the question is asking you to calculate a confidence interval for the population mean aluminum content of the cans produced at the Akron plant. To do this, you will need the data from Column A, which contains the aluminum content data from a sample of 100 cans.
a) To construct a 95% confidence interval for the population mean aluminum content, you will need to calculate the sample mean, sample standard deviation, and the critical value.
1. Calculate the sample mean (x̄) by summing up all the aluminum content values in Column A and dividing by the sample size (100).
2. Calculate the sample standard deviation (s) using the formula: s = sqrt(Σ(x - x̄)^2 / (n - 1)), where Σ represents the sum of the squared differences between each aluminum content value (x) and the sample mean (x̄), n is the sample size (100), and sqrt() denotes the square root function.
3. Determine the critical value (t) corresponding to a 95% confidence level. The critical value is based on the t-distribution and depends on the sample size. You can look up the critical value for a 95% confidence level in a t-distribution table using n-1 degrees of freedom (n-1 = 99 in this case).
4. Calculate the margin of error (ME) using the formula: ME = t * (s / sqrt(n)), where t is the critical value, s is the sample standard deviation, and sqrt() denotes the square root function.
5. Finally, calculate the confidence interval by subtracting the margin of error from the sample mean (x̄) to get the lower bound, and adding the margin of error to the sample mean (x̄) to get the upper bound.
b) The extra credit part of the question asks you to construct a 95% confidence interval for the total expenditure on aluminum at the Akron plant. To do this, you will need to convert the sample mean (x̄) and sample standard deviation (s) obtained in part (a) from grams per can to dollars per can.
1. Convert the sample mean (x̄) from grams per can to pounds per can by dividing it by 454 (since there are 454 grams in one pound).
2. Convert the sample standard deviation (s) from grams per can to pounds per can using the same conversion factor (divide by 454).
3. Multiply the converted sample mean (in pounds per can) by the market price for aluminum ($0.95 per pound) to get the total expenditure per can in dollars.
4. Calculate the margin of error (ME) using the formula: ME = t * (s / sqrt(n)), where t is the critical value, s is the converted sample standard deviation (in pounds per can), and sqrt() denotes the square root function.
5. Calculate the confidence interval by subtracting the margin of error from the total expenditure per can to get the lower bound, and adding the margin of error to the total expenditure per can to get the upper bound.
c) Based on your answers in part (a) and part (b), you can interpret the results as follows:
- The confidence interval for the population mean aluminum content (in grams per can) tells you that with 95% confidence, the true mean aluminum content of all the cans produced at the Akron plant falls within the calculated range.
- The confidence interval for the total expenditure on aluminum (in dollars per can) tells you that with 95% confidence, the true average expenditure on aluminum for all the cans produced at the Akron plant falls within the calculated range. If the cans were produced at the target aluminum content (15 grams), AluminiCorp could potentially save money by reducing the total expenditure on aluminum per can.