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## Homework 4

Due date: Thursday, Nov. 16, 2017.

Note: all hypothesis testing problems should include the null and alternative hypotheses and report the p-value of the data.

1. A company is said to be out of compliance if more than 7% of all invoices contain errors, and it is said to be seriously out of compliance if more than 10% of all invoices contain errors. Suppose an auditor randomly selects a sample of 1000 invoices and finds that 120 contained errors.
• Construct a 95% confidence interval for this company's error rate.
• How should the company be rated if statements about being out of compliance or seriously out of compliance require 5% level of significance?
• What is the probability a company would be rated as seriously out of compliance by this test if 15% of all invoices at that company contain errors?
• What sample size should the auditor use to estimate the error rate to within 2% with 99% confidence if it is assumed that the error rate will be no more than 15%?
• Suppose the 120 erroneous invoices can be treated as a random sample from the population of all erroneous invoices. The error amounts are contained in the file
http://www.UTDallas.edu/~ammann/stat3355scripts/Invoice1.txt
Note: since this file just contains a single set of numeric values, you can use the scan() function in R to read this data. For example,
```InvErr = scan("http://www.UTDallas.edu/~ammann/stat3355scripts/Invoice1.txt")
```
Construct a 95% confidence interval for the mean error amount. Also obtain and interpret a quantile-quantile plot of these invoice errors compared to the normal distribution.

2. A large corporation has a plant that has started a pilot program to give stock options for its assembly line workers as part of their benefits package. The corporation would like to determine if the mean quality score for this plant exceeds the current average quality measure of 82. A random sample of 25 production units is selected from this plant and the quality scores for these units are obtained. The sample mean score for these units is 86.2 with a s.d. of 10.4.
• Does this plant have a higher mean score at the 5% level of significance?
• Construct a 90% confidence interval for the mean score of this plant.
• Use this data as a preliminary sample to determine the sample size required to estimate the mean quality score under this program to within 2.0 with 95% confidence.

3. A random sample of 48 students took an SAT preparation course prior to taking the SAT. The sample mean of their quantitative SAT scores was 560 with a s.d. of 80, and the sample mean of their verbal SAT scores was 520 with a s.d. of 110.
• Construct 95% confidence intervals for the mean quantitative SAT and the mean verbal SAT scores of all students who take this course.
• Construct 95% confidence intervals for the standard deviations of the QSAT and VSAT scores of all students who take this course.
• What sample size would be needed to estimate the mean VSAT score with 95% confidence and with error of no more than 15 if it is assumed that the s.d. is no more than 100?
• Suppose the mean scores for all students who took the SAT at that time was 545 for the quantitative and 480 for the verbal? Do the means for students who take this course differ from the means for all students at the 10% level of significance?

4. A fabrication plant has just completed a contract to supply memory chips to a computer manufacturer that requires the defective rate of these chips to be no more than 6%. The plant has just installed new equipment to produce these chips. An initial production run of 500 chips will be obtained and one of three actions will be taken depending on the results of this run. If it is shown that more than 6% of the chips are defective, then the equipment will be recalibrated to reduce the defective rate; if it is shown that fewer than 6% of chips are defective, then cheaper raw material will be used to reduce costs; otherwise, the equipment will be unchanged and production will begin. Suppose that among the initial production of 500 chips it was found that 40 chips were defective.
• What action should plant management take at the 5% level of significance?
• What is the probability that the null hypothesis would be rejected if the population defective rate is actually 10%?
• Estimate the current defective rate with a 95% confidence interval.
• In the future plant management would like to estimate the defective rate to within 2% using 95% confidence intervals. What sample size would be required to accomplish this if it is assumed that the defective rate would be no more than 10%?

Next: Summary of Methods Up: Homework and Project Assignments Previous: Homework 3
ammann
2017-11-16