We need to use the signal from a piezoelectric vibration sensor in a computer-aided instrumentation system. The signal is known to contain components with frequencies up to 30 kHz. What is the minimum sampling rate that should be specified? Suppose that we want the resolution of the sampled values to be 0.1 percent (or better) of the full range of the ADC. What is the fewest number of bits that should be specified for the ADC?

A dissolved oxygen analyzer in a bioreactor is used to provide composition measurements at each sampling time in a feedback control loop. The open-loop transfer function is given by

$$\begin{aligned} G_{O L} & =G_c G \\ G_c & =2\left(1+\frac{1}{8 s}\right) \quad G=\left(\frac{10}{12 s+1}\right) e^{-2 s} \end{aligned}$$

(a) Suppose that a sampling period of $\Delta t=1 \mathrm{~min}$ is selected. Derive $G_c(z)$ using a backward-difference approximation of $G_c(s)$. (b) If a unit step change in the controller error signal $e(t)$ is made, calculate the sampled open-loop response $y_m(k \Delta t)$ using simulation with a zero-order hold after the controller $G_c$ and before the process $G$ in the block diagram constructed via Simulink.

Decide whether you should use a normal sampling distribution or a t-sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim.

An administrator of a state university system says that the mean time full-time faculty spend with students outside of class is 337 hours per week. (The hours were counted for each student, so a professor 's spending five hours with a group of four students would count as 20 hours.) The mean time a random sample of 50 full-time faculty members spent with students outside of class was 332 hours per week and the standard deviation was 10 hours per week. At $\alpha$ = 0.01, test the administrator 's claim.

Electronics Inc.'s human resources division would like to add a dental plan to the list of benefits. How much does the average employee and his or her family spend on dental care annually? The average expense last year was $1,820, with a standard deviation of$660, according to a sampling of 45 employees. With the help of this data, construct a 95% confidence interval for the population mean.

Independent random samples of 64 observations each are chosen from two normal populations with the following means and standard deviations:

Let $\bar{x}_1$ and $\bar{x}_2$ denote the two sample means.

Give the mean and standard deviation of the sampling distribution of $\bar{x}_2$.

Any time we conduct a survey, we must take care to avoid undercoverage. Suppose we plan to select 500 names from he city phone book, call their homes between noon and 4 P.M., and interview whoever answers, anticipating contacts with at lea t 200 people.

Describe a more convenient, but still random, sampling strategy.

Some charged particles are suspended inside a hollow metal object by supports made of a material that is an electrical insulator. The metal object has charge +q distributed over its outer surface and charge -2q distributed over its inner surface. (a) How much charge is suspended in the cavity? (b) What is the charge on the object?

Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither:

Claim about daily rainfall amounts in Boston: $\mu<0.20$ in. Sample data: $n=19$, $\bar{x}=0.10 \mathrm{in}, s=0.26 \mathrm{in}$. The sample data appear to come from a population with a distribution that is very far from normal, and $\sigma$ is unknown.

A company packaging snack foods maintains quality control by randomly choosing 10 cases from each day's production and weighing the bags. Then they open one bag from each case and inspect the contents.

Identify the next items (if feasible). If you can't describe, then say so-this often happens when we read about a survey.

c) The sampling frame

Here is the population of all five U.S. presidents who had professions in the military, along with their age at inauguration: Eisenhower (62), Grant (46), Harrison (68), Taylor (64), and Washington (57). Assume that all samples are selected without replacement. a. After listing all of the possible samples of size n = 2, find the mean and standard deviation of the sample means. b. After listing all of the possible samples of size n = 3, find the mean and standard deviation of the sample means. c. After listing all of the possible samples of size n = 4, find the mean and standard deviation of the sample means. d. When sampling without replacement, do sample means tend to target the value of the population mean? e. Based on the preceding results, how is the variation of the sampling distribution of sample means affected by increasing sample size?

When the statistic that estimates an unknown parameter has a normal distribution, a confidence interval for the parameter has the form

estimate $\pm z^* \sigma_{\text {estimate }}$

In a complex sample survey design, the estimate of the population mean and the standard deviation of this estimate require elaborate computations. But when we are given the estimate and its standard deviation, we can calculate a confidence interval for $\mu$ without knowing the formulas that led to the numbers given.

A report based on the Current Population Survey estimates the median weekly earnings of families of wage or salary workers as $664 and also estimates that the standard deviation of this estimate is$3.50. The Current Population Survey uses an elaborate multistage sampling design to select a sample of about 60,000 households. The sampling distribution of the estimated median income is approximately normal. Give a 95% confidence interval for the median weekly earnings of all families of wage and salary workers.

The cost of performing inspection sampling moves inversely to the cost of quality failures. We can reduce the cost of quality failures by increased levels of inspection, but that of course would increase the cost of inspection. Can you think of any methods to reduce the cost of quality failures without increasing a company's cost of inspection? Think specifically in terms of material purchased from vendors.

The registrar keeps an alphabetical list of all undergraduates, with their current addresses. Suppose there are 10,000 undergraduates in the current term. Someone proposes to choose a number at random from 1 to 100, count that far down the list, taking that name and every 100th name after it for the sample. a) Is this a probability method? b) Is it the same as simple random sampling? c) Is there selection bias in this method of drawing a sample?

Open the applet Sampling Distributions. On the pull-down menu to the right of the top graph, select Uniform. The box to the left of the top graph displays the population mean, median, and standard deviation of the original distribution. Run the applet for the sample size $n=30$ and the number of samples $N=1,000$. Observe the shape of the graph of the sample means and record the mean, median, and standard deviation of the sample means.