Consider the gain estimation problem in Exercise 8-4.(a)

Dairy cows at large commercial farms often receive injections of bST (Bovine Somatotropin), a hormone used to spur milk production. Bauman et al. (Journal of Dairy Science, 1989) reported that 12 cows given bST produced an average of 28.0 kg/d of milk. Assume that the standard deviation of milk production is 2.25 kg/d. (a) Find a 99% confidence interval for the true mean milk production. (b) If the farms want the confidence interval to be no wider than \(\pm 1.25 \mathrm{~kg} / \mathrm{d}\), what level of confidence would they need to use? Equation Transcription: Text Transcription: +/-1.25 kg/d

For a normal population with known variance \(\sigma^{2}\), answer the following questions: (a) What is the confidence level for the interval \(\bar{x}-2.14 \sigma / \sqrt{n} \leq \mu \leq \bar{x}+2.14 \sigma / \sqrt{n}\)? (b) What is the confidence level for the interval \(\bar{x}-2.49 \sigma / \sqrt{n} \bar{x}-2.49 \sigma / \sqrt{n} \leq \mu \leq \bar{x}+2.49 \sigma / \sqrt{n}\)? (c) What is the confidence level for the interval \(\bar{x}-1.85 \sigma / \sqrt{n} \leq \mu \leq \bar{x}+1.85 \sigma / \sqrt{n}\)? (d) What is the confidence level for the interval \(\mu \leq \bar{x}+2.00 \sigma / \sqrt{n}\)? (e) What is the confidence level for the interval \(\bar{x}-1.96 \sigma / \sqrt{n} \leq \mu\)? Equation Transcription: Text Transcription: sigma^2 x bar-2.14sigma/sqrt n </= mu </= x bar +2.14sigma/sqrt n x bar-2.49sigma/sqrt n x bar-2.49sigma/sqrt n </= mu </= x bar +2.49/sqrt n x bar-1.85sigma/sqrt n </= mu </= x+1.85sigma/sqrt n mu </= x bar+2.00 sigma/sqrt n x bar-1.96sigma/sqrt n </= mu

A confidence interval estimate is desired for the gain in a circuit on a semiconductor device. Assume that gain is normally distributed with standard deviation \(\mathrm{s}=20\). (a) Find a 95% CI for m when \(n=10\) and \(\bar{x}=1000\). (b) Find a 95% CI for m when \(n=25\) and \(\bar{x}=1000\). (c) Find a 99% CI for m when \(n=10\) and \(\bar{x}=1000\). (d) Find a 99% CI for m when \(n=25\) and \(\bar{x}=1000\). (e) How does the length of the CIs computed change with the changes in sample size and conidence level? Equation Transcription: Text Transcription: s=20 n=10 x bar=1000 n=25 x bar=1000 n=10 x bar=1000 n=25 x bar=1000

Consider the one-sided confidence interval expressions for a mean of a normal population. (a) What value of \(z_{\alpha}\) would result in a 90% CI? (b) What value of \(z_{\alpha}\) would result in a 95% CI? (c) What value of \(z_{\alpha}\) would result in a 99% CI?

For a normal population with known variance \(\sigma^2\) : (a) What value of \(z_{\alpha / 2}\) in Equation 8-5 gives 98% confidence? (b) What value of \(z_{\alpha / 2}\) in Equation 8-5 gives 80% confidence? (c) What value of \(z_{\alpha / 2}\) in Equation 8-5 gives 75% confidence?

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02, 61.98) and (39.95, 60.05) (a) What is the value of the sample mean? (b) One of these intervals is a 95% CI and the other is a 90% CI. Which one is the 95% CI and why?

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (37.53, 49.87) and (35.59, 51.81) (a) What is the value of the sample mean? (b) One of these intervals is a 99% CI and the other is a 95% CI. Which one is the 95% CI and why?

Following are two confidence interval estimates of the mean m of the cycles to failure of an automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate the failure). \(3124.9 \leq \mu \leq 3215.7\) \(3110.5 \leq \mu \leq 3230.1\) (a) What is the value of the sample mean cycles to failure? (b) The conidence level for one of these CIs is 95% and for the other is 99%. Both CIs are calculated from the same sample data. Which is the 95% CI? Explain why. Equation Transcription: Text Transcription: 3124.9 </= mu </= 3215.7 3110.5 </= mu </= 3230.1

The yield of a chemical process is being studied. From previous experience, yield is known to be normally distributed and . The past five days of plant operation have resulted in the following percent yields: 91.6, 88.75, 90.8, 89.95, and 91.3. Find a 95% two-sided confidence interval on the true mean yield.

Past experience has indicated that the breaking strength of yarn used in manufacturing drapery material is normally distributed and that . random sample of nine specimens is tested, and the average breaking strength is found to be 98 psi. Find a 95% two-sided confidence interval on the true mean breaking strength

Suppose that \(n=100\) random samples of water from a freshwater lake were taken and the calcium concentration (milligrams per liter) measured. A 95% CI on the mean calcium concentration is \(0.49 \leq \mu \leq 0.82\). (a) Would a 99% CI calculated from the same sample data be longer or shorter? (b) Consider the following statement: There is a 95% chance that \(\mu\) is between 0.49 and 0.82. Is this statement correct? Explain your answer. (c) Consider the following statement: If \(n=100\) random sam-ples of water from the lake were taken and the 95% CI on \(\mu\) computed, and this process were repeated 1000 times, 950 of the CIs would contain the true value of \(\mu\). Is this statement correct? Explain your answer. Equation Transcription: Text Transcription: n=100 0.49</=mu</=0.82 mu n=100 mu

Consider the gain estimation problem in Exercise 8-4. (a) How large must n be if the length of the 95% CI is to be 40? (b) How large must n be if the length of the 99% CI is to be 40?

The diameter of holes for a cable harness is known to have a normal distribution with \(\sigma=0.01\) inch. A random sample of size 10 yields an average diameter of 1.5045 inch. Find a 99% two-sided confidence interval on the mean hole diameter.

A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with \(\sigma=0.001\) millimeters. A random sample of 15 rings has a mean diameter of \(\bar{x}=74.036\) millimeters. (a) Construct a 99% two-sided confidence interval on the mean piston ring diameter. (b) Construct a 99% lower-confidence bound on the mean piston ring diameter. Compare the lower bound of this confi-dence interval with the one in part (a). Equation Transcription: Text Transcription: sigma=0.001 x bar=74.036

A civil engineer is analyzing the compressive strength of concrete. Compressive strength is normally distributed with \(\sigma^{2}=1000(\mathrm{psi})^{2}\). A random sample of 12 specimens has a mean compressive strength of \(\bar{x}=3250\) psi. (a) Construct a 95% two-sided confidence interval on mean compressive strength. (b) Construct a 99% two-sided confidence interval on mean compressive strength. Compare the width of this confidence interval with the width of the one found in part (a). Equation Transcription: Text Transcription: sigma^2=1000(psi)^2 x bar=3250

The life in hours of a 75-watt light bulb is known to be normally distributed with \(\sigma=25\) hours. A random sample of 20 bulbs has a mean life of \(\bar{x}=1014\) hours. (a) Construct a 95% two-sided confidence interval on the mean life. (b) Construct a 95% lower-confidence bound on the mean life. Compare the lower bound of this confidence interval with the one in part (a). Equation Transcription: Text Transcription: sigma^2=25 x bar=1014

Suppose that in Exercise 8-14 we wanted the error in estimating the mean life from the two-sided confidence interval to be five hours at 95% confidence. What sample size should be used?

Problem 19E By how much must the sample size n be increased if the length of the CI on µ in Equation 8-5 is to be halved?

Suppose that in Exercise 8-14 you wanted the total width of the two-sided confidence interval on mean life to be six hours at 95% confidence. What sample size should be used?

Suppose that in Exercise 8-15 it is desired to estimate the compressive strength with an error that is less than 15 psi at 99% confidence. What sample size is required?

If the sample size n is doubled, by how much is the length of the CI on \(\mu\) in Equation 8-5 reduced? What happens to the length of the interval if the sample size is increased by a factor of four?

An article in the Journal of Agricultural Science [“The Use of Residual Maximum Likelihood to Model Grain Quality Characteristics of Wheat with Variety, Climatic and Nitrogen Fertilizer Effects” (1997, Vol. 128, pp. 135–142)] investigated means of wheat grain crude protein content \(\text { (CP) }\) and Hagberg falling number \(\text { (HFN) }\) surveyed in the United Kingdom. The analysis used a variety of nitrogen fertilizer applications \(\text { (kg N/ha) }\), temperature \(\left({ }^{\circ} \mathrm{C}\right)\), and total monthly rainfall \(\text { mm) }\). The following data below describe temperatures for wheat grown at Harper Adams Agricultural College between 1982 and 1993. The temperatures measured in June were obtained as follows: Assume that the standard deviation is known to be \(\sigma=0.5\). (a) Construct a \(99 \%\) two-sided confidence interval on the mean temperature. (b) Construct a \(95 \%\) lower-confidence bound on the mean temperature. (c) Suppose that you wanted to be \(95 \%\) confident that the error in estimating the mean temperature is less than 2 degrees Celsius. What sample size should be used? (d) Suppose that you wanted the total width of the two-sided confidence interval on mean temperature to be 1.5 degrees Celsius at \(95 \%\) confidence. What sample size should be used? Equation Transcription: Text Transcription: (CP) (HFN) (kg N/ha) (°C) (mm) \sigma = 0.5 99% 95% 95% 95%

Problem 22E Ishikawa et al. (Journal of Bioscience and Bioengineering, 2012) studied the adhesion of various biofilms to solid surfaces for possible use in environmental technologies. Adhesion assay is conducted by measuring absorbance at A500. Suppose that for the bacterial strain Acinetobacter, five measurements gave readings of 2.69, 5.76, 2.67, 1.62 and 4.12 dyne-cm2. Assume that the standard deviation is known to be 0.66 dyne-cm2. (a) Find a 95% confidence interval for the mean adhesion. (b) If the scientists want the confidence interval to be no wider than 0.55 dyne-cm2, how many observations should they take?