To use the t -statistic to test for a difference between the means of two populations, what assumptions must be made about the two populations? About the two samples?

A computer intrusion detection system (IDS) is designed to provide an alarm whenever someone intrudes (e.g., through nauthorized access) into a computer system. A robabilistic evaluation of a system with two independently operating intrusion detection systems (a double IDS) was published in the Journal of Research of the National Institute of Standards and Technology (Nov.–Dec. 2003). Consider a double IDS with system A and system B. If there is an intruder, system A sounds an alarm with probability .9 and system B sounds an alarm with probability .95. If there is no intruder, the probability that system A sounds an alarm (i.e., a false alarm) is .2 and the probability that system B sounds an alarm is .1. a. Use symbols to express the four probabilities just given. b. If there is an intruder, what is the robability that both systems sound an alarm? c. If there is no intruder, what is the probability that both systems sound an alarm? d. Given that there is an intruder, what is the probability that at least one of the systems sounds an alarm?

Suppose you conduct a t -test for the null hypothesis

$$H _ { 0 } : \mu = 1,000$$

versus the alternative hypothesis

$$H _ { a } : \mu > 1,000$$

based on a sample of 17 observations. The test results are

$$t = 1.89 , p-value = .038$$

What assumptions are necessary for the validity of this procedure?

Disaggregation of annually reported accounting data. Researchers have constructed a new measure of disclosure quality-called disaggregation quality $(D Q)$-based on the level of disaggregation of financial data items in a firm's annual reports (Journal of Accounting Research, December 2015). Quantitative DQ values were computed for each of 125,873 firms that had annual balance sheet data from 1973 to 2011 reported on Compustat. One objective is to describe the distribution of the $\mathrm{DQ}$ values for firms reported on Compustat. What is the variable measured in this study?

The equation

$$p = 0.4 t + 10.5$$

approximates the average hourly pay p (in dollars) of a U.S. manufacturing worker, t years after 1990. a. Graph the equation. b. What information can be obtained from the p-intercept of the graph? c. Suppose the current trend continues. From the graph, estimate the average hourly pay for manufacturing workers in 2014.

The following is a 90% confidence interval for p : (.26, .54). How large was the sample used to construct this interval?

In a paired difference experiment, when should the observations be paired, before or after the data are collected?

A computer salesman finds that if he sells n computers, he can make a profit of $(200 − 4n) per computer. How many computers should he sell in order to maximise his profit?

An experiment reported in Popular Science compared fuel economies for two types of similarly equipped diesel mini-trucks. Let us suppose that 12 Volkswagen and 10 Toyota trucks were tested in 90kilometer-per-hour steady-paced trials. If the 12 Volkswagen trucks averaged 16 kilometers per liter with a standard deviation of 1.0 kilometer per liter and the 10 Toyota trucks averaged 11 kilometers per liter with a standard deviation of 0.8 kilometer per liter, construct a 90% confidence interval for the difference between the average kilometers per liter for these two mini-trucks. Assume that the distances per liter for the truck models are approximately normally distributed with equal variances.

When bonding teeth, orthodontists must maintain a dry field. A new bonding adhesive (called “Smartbond”) has been developed to eliminate the necessity of a dry field. However, there is concern that the new bonding adhesive is not as strong as the current standard, a composite adhesive. ( Trends in Biomaterials & Artificial Organs , Jan. 2003.) Tests on a sample of 10 extracted teeth bonded with the new adhesive resulted in a mean breaking strength (after 24 hours) of

$$\overline { x } = 5.07$$

Mpa and a standard deviation of s = .46 Mpa, where Mpa = megapascal, a measure of force per unit area. Orthodontists want to know if the true mean breaking strength of the new bonding adhesive is less than 5.70 Mpa, the mean breaking strength of the composite adhesive. Conduct the appropriate analysis for the orthodontists.

$$\text { Use } \alpha = .01$$

Refer to the international Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots. The researchers obtained a random sample of 106 social robots through a Web search and determined the number that were designed with legs, but no wheels. Let $\hat{p}$ represent the sample proportion of social robots designed with legs, but no wheels. Assume that in the population of all social robots, 40% are designed with legs, but no wheels. Give the mean and standard deviation of the sampling distribution of $\hat{p}$.

Suppose you want to estimate the difference between two population means correct to within 2.2 with probability .95. If prior information suggests that the population variances are approximately equal to

$$\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } = 15$$

and you want to select independent random samples of equal size from the populations, how large should the sample sizes,

$$n_1 \ and \ n_2,$$

be?

Find the reciprocal of -2/5.

Refer to the Business and Society (Mar. 2011) study on the sustainability behaviors of CPA corporations. Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each in a sample of 992 senior managers at CPA firms. The data (where higher point values indicate a higher level of support for sustainability) are saved in the referenced file. The CEO of a CPA firm claims that the true mean level of support for sustainability is 75 points. Specify the null and alternative hypotheses for testing this claim. One Sample Statistics $$ \begin{matrix} \text{ } & \text{N} & \text{Mean} & \text{Std. Deviation} & \text{Std. Error Mean}\\ \text{SUPPORT} & \text{992} & \text{67.76} & \text{26.871} & \text{.853}\\ \end{matrix} $$ One sample Test. Test Value = 75. $$ \begin{matrix} \text{ } & \text{t} & \text{df} & \text{Sig. (2-talled)} & \text{Mean Dífference} & \text{95\\% Contidence lnterval of the Difference}\\ \text{SUPPORT} & \text{-8.492} & \text{991} & \text{.000} & \text{-7.245} & \text{Lower -8.92} & \text{Upper -5.57}\\ \end{matrix} $$