Solution Found!
A chemist is calibrating a spectrophotometer that will be
Chapter 7, Problem 5SE(choose chapter or problem)
A chemist is calibrating a spectrophotometer that will be used to measure the concentration of carbon monoxide (CO) in atmospheric samples. To check the calibration, samples of known concentration are measured. The true concentrations \((x)\) and the measured concentrations \((y)\) are given in the following table. Because of random error, repeated measurements on the same sample will vary. The machine is considered to be in calibration if its mean response is equal to the true concentration.
To check the calibration, the linear model \(y=\beta_{0}+\beta_{1} x+\varepsilon\) is fit. Ideally, the value of \(\beta_{0}\) should be 0 and the value of \(\beta_{1}\) should be 1.
a. Compute the least-squares estimates \(\widehat{\beta}_{0}\) and \(\widehat{\beta}_{1}\).
b. Can you reject the null hypothesis \(H_{0}: \beta_{0}=0\)?
c. Can you reject the null hypothesis \(H_{0}: \beta_{1}=1\)?
d. Do the data provide sufficient evidence to conclude that the machine is out of calibration?
e. Compute a 95% confidence interval for the mean measurement when the true concentration is 20 ppm.
f. Compute a 95% confidence interval for the mean measurement when the true concentration is 80 ppm.
g. Someone claims that the machine is in calibration for concentrations near 20 ppm. Do these data provide sufficient evidence for you to conclude that this claim is false? Explain.
Questions & Answers
QUESTION:
A chemist is calibrating a spectrophotometer that will be used to measure the concentration of carbon monoxide (CO) in atmospheric samples. To check the calibration, samples of known concentration are measured. The true concentrations \((x)\) and the measured concentrations \((y)\) are given in the following table. Because of random error, repeated measurements on the same sample will vary. The machine is considered to be in calibration if its mean response is equal to the true concentration.
To check the calibration, the linear model \(y=\beta_{0}+\beta_{1} x+\varepsilon\) is fit. Ideally, the value of \(\beta_{0}\) should be 0 and the value of \(\beta_{1}\) should be 1.
a. Compute the least-squares estimates \(\widehat{\beta}_{0}\) and \(\widehat{\beta}_{1}\).
b. Can you reject the null hypothesis \(H_{0}: \beta_{0}=0\)?
c. Can you reject the null hypothesis \(H_{0}: \beta_{1}=1\)?
d. Do the data provide sufficient evidence to conclude that the machine is out of calibration?
e. Compute a 95% confidence interval for the mean measurement when the true concentration is 20 ppm.
f. Compute a 95% confidence interval for the mean measurement when the true concentration is 80 ppm.
g. Someone claims that the machine is in calibration for concentrations near 20 ppm. Do these data provide sufficient evidence for you to conclude that this claim is false? Explain.
ANSWER:Step 1 of 9
A confidence interval is a statistical tool used to estimate the range within which we believe a true population parameter lies. It provides a range of values around a sample statistic, such as the mean or proportion, and expresses the level of confidence that the true parameter falls within that range. In other words, it quantifies the uncertainty associated with our estimate.