Question 1: (by hand)
A major manufacturing firm producing PCB for electrical insulation discharges small amounts
from the plant. We assume that the amount of PCB discharge per water specimen is normally
distributed. Production will be halted if there is evidence that the mean PCB amount discharged
in the water exceeds 3 ppm (parts per million). A random sample of 16 water specimens
produced X = 3.2 ppm and a sample standard deviation S = 0.3
a) Do these statistics provide sufficient evidence to halt the process? Use α = 0.05 .
b) Briefly discuss the consequences of type I and type II errors. From your point of view, which
of the two errors is the most serious?
c) Construct a 90% confidence interval for the true mean PCB amount discharged in the water.
Question 2: (R)
The design of controls and instruments has a large effect on how easily people can use them.
A student project investigated this effect by asking 25 right-handed students to turn a
knob (with their right hands) that moved an indicator by screw action. There were two
identical instruments, one with a right-hand thread (knob turns clockwise) and the other
with a left-hand thread (knob turns counterclockwise). The data gives the times required
to move the indicator a fixed distance.
a) Each of the 25 students used both instruments. Explain how the experiment should be
arranged and how randomization should be done. Think about whether any learning is
involved that might make it easier to do the second task of turning a knob.
b) Make a histogram, a boxplot, and a QQ plot of the paired differences. Check skewness,
ouliers, and departure from normality linearity supports normality in the QQ plot). Report
c) The project hoped to show that right-handed people find right handed threads easier to use.
State the appropriate null and alternative hypotheses about the mean time required and
carry out the test. Base yourself on the R output to report tht value of the test statistic and
the P value. Write your conclusion in terms specific to the setting (stay away from the
formula “I reject the null hypothesis”...). Include also a 90% confidence interval for the
difference between the two mean times.
d) Use b) to check the assumption that the paired differences come from a normal distribution.
Call Rcmdr with library(Rcmdr), then open the file with Data → Import data → from text file ...
Use Commas as the field separator’.
b) This is a matched pairs design. To find the paired differences choose
o Data → Manage variables in active data sets → Compute new variable
o Type Difference in the “New variable name” box
o Type Left.Thread – Right.Thread in the “Expression to compute box” and click OK
o If you click on the view data set button, you will see the Difference column.
c) Obtain histogram, boxplot, and QQ plot of the Difference variable by doing consecutively
- Graphs → Histogram → etc.
- Graphs → Boxplot → etc.
- Graphs → Quantile-comparison → etc.
The paired t-test and the confidence interval
o Statistics → Means → Paired t-test
o Pick the relevant first and second variables
o From the Options tab
- Leave the alternative hypothesis at the two-sided state
- Enter the confidence level (in decimal expansion)
- Copy and paste the output
o You will easily locate the confidence bounds in the output, and thus your answer to part
b). For the statistical test, read question c) carefully. If the test is two-sided, you have all
you need to answer d). If the test is one-sided, then the P value of the one-sided test is
equal to half the P value of the two-sided test (this must be clear to you; make sure the
mean of the paired differences reflects the direction of the alternative hypothesis.) Ready
to answer c)!