Question

The laser sight Jupiter uses for surveying is a little off. The mean error is 0.29m, meaning that it tends to provide measurements that are 0.29m too long. The standard deviation of the errors is 0.35m. She decides to recalibrate the device, but she wants to test it afterward to see if she made things better or worse. She collects a random sample of 43 measurements of a 100m distance which is found in the data file “surveying.xlsx” on MyLab.

Survery.xlsx

Measurements

99.87 100.17 100.9 100.65 99.84 100.59 99.62 99.09 99.43

100.13 99.74 99.21 100.3 99.98 100.37 100.18 99.83 100.34

100.3 100.1 100.11 99.47 100.48 101.05 99.61 99.93 100.74

99.99 100.06 100.55 99.85 99.73 99.93 100.6 99.89 100.37

99.78 100.89 100.29 99.91 100.42 100.11 99.66

a. Identify the population of interest.

b. Identify the variable of interest. What type of variable is it?

c. If she measured the 100m distance BEFORE recalibrating, what would the mean of the measurements have been?

d. If she wishes to assess how far off the sight is AFTER recalibrating, what parameter should she estimate?

e. Are the conditions for estimating the parameter you chose in part d met? What assumptions would you need to make?

f. Estimate the parameter you chose in part d with 99% confidence. Does her recalibration appear to have improved this situation?

g. If she wishes to assess how reliable the sight is AFTER recalibrating, what parameter should she estimate?

h. Are the conditions for estimating the parameter you chose in part g met? What assumptions would you need to make?

i. Estimate the parameter you chose in part g with 99% confidence. Does her recalibration appear to have improved this situation?

j. Overall, do you think her recalibration made things better or worse?

Answer 1

a. Population: 100m distance measured by the laser sight Jupiter after recalibrating

b. The variable of interest: Measurement by the laser sight Jupiter after recalibrating. Type: Continuous variable.

c. If she measured the 100m distance BEFORE recalibrating, the mean of the measurements=100+0.29=100.29.

d. Parameter of interest=Population mean error measurement by the laser sight Jupiter after recalibrating=.

e. Here we assume that the errors come from normal distribution.

f. 99% CI for : (-0.0881, 0.2769).

R code:

x=c(99.87, 100.17, 100.9, 100.65, 99.84, 100.59, 99.62, 99.09, 99.43,100.13,
99.74, 99.21, 100.3, 99.98, 100.37, 100.18, 99.83, 100.34,100.3, 100.1, 100.11,
99.47, 100.48, 101.05, 99.61, 99.93, 100.74,99.99, 100.06, 100.55, 99.85, 99.73,
99.93, 100.6, 99.89, 100.37,99.78, 100.89, 100.29,99.91, 100.42, 100.11, 99.66)-100
n=length(x)
m=mean(x)
s=sd(x)
LB=m-qt(0.995,n-1)*s/sqrt(n)
UB=m+qt(0.995,n-1)*s/sqrt(n)
round(LB,4)
round(UB,4)

Yes, since upper bound of CI is less than 100.29.

g. Sample standard deviation of error.

h. The errors come from normal distribution.

i. 99% CI for population standard deviation of error: (0.1192, 0.3732).

R code:

x=c(99.87, 100.17, 100.9, 100.65, 99.84, 100.59, 99.62, 99.09, 99.43,100.13,
99.74, 99.21, 100.3, 99.98, 100.37, 100.18, 99.83, 100.34,100.3, 100.1, 100.11,
99.47, 100.48, 101.05, 99.61, 99.93, 100.74,99.99, 100.06, 100.55, 99.85, 99.73,
99.93, 100.6, 99.89, 100.37,99.78, 100.89, 100.29,99.91, 100.42, 100.11, 99.66)-100
n=length(x)
s=sd(x)
LB=(n-1)*s^2/qchisq(0.995,n-1)
UB=(n-1)*s^2/qchisq(0.005,n-1)

round(LB,4)
round(UB,4)

Since CI for population standard deviation of error contains 0.35, we are 99% confident that population sd of error is not significantly different from 0.29.

J. Since population standard deviation of errors after recalibration is unchanged but population mean of errors after recalibration is low, we think that her recalibration made things better.