True or False? The normal distribution can be used to approximate a binomial distribution when $np$ is less than $5$.

Consider the fire-fighting foam expanding agents investigated, in which five observations of each agent were recorded. Suppose that, if agent 1 produces a mean expansion that differs from the mean expansion of agent 2 by 1.5 , we would like to reject the null hypothesis with probability at least 0.95 . (a) What sample size is required? (b) Do you think that the original sample size in Exercise 5-18 was appropriate to detect this difference? Explain your answer.

An article appeared in The Wall Street Journal on Tuesday, April 27, 2010, with the title “Eating Chocolate Is Linked to Depression.” The article reported on a study funded by the National Heart, Lung and Blood Institute (part of the National Institutes of Health) and the University of California, San Diego, that examined 931 adults who were not taking antidepressants and did not have known cardiovascular disease or diabetes. The group was about 70% men and the average age of the group was reported to be about 58. The participants were asked about chocolate consumption and then screened for depression using a questionnaire. People who score less than 16 on the questionnaire are not considered depressed and those with scores above 16 and less than or equal to 22 are considered possibly depressed and those with scores above 22 are considered likely to be depressed. The survey found that people who were not depressed ate an average of 5.4 servings of chocolate per month, possibly depressed individuals ate an average of 8.4 servings of chocolate per month, while those individuals who scored above 22 and were likely to be depressed ate the most chocolate, an average of 11.8 servings per month. No differentiation was made between dark and milk chocolate. Other foods were also examined, but no pattern emerged between other foods and depression. Does this type of study establish a cause-and-effect link between chocolate consumption and depression? How would the study have to be conducted to establish such a cause-and effect link?

An article in Experimental Brain Research ["Synapses in the Granule Cell Layer of the Rat Dentate Gyrus: Serial-Sectioning Study" ] showed the ratio between the numbers of symmetrical and total synapses on somata and azon initial segments of reconstructed granule cells in the dentate gyrus of a 12-week-old rat:

$$\begin{array}{llllll}0.65 & 0.90 & 0.78 & 0.94 & 0.40 & 0.94 \\ 0.91 & 0.86 & 0.53 & 0.84 & 0.42 & 0.50 \\ 0.50 & 0.68 & 1.00 & 0.57 & 1.00 & 1.00 \\ 0.84 & 0.9 & 0.91 & 0.92 & 0.96 & \\ 0.96 & 0.56 & 0.67 & 0.96 & 0.52 & \\ 0.89 & 0.60 & 0.54 & & & \end{array}$$

Use the data to test $H_0: \sigma^2=0.02$ versus $H_1: \sigma^2 \neq 0.02$ using $\alpha=0.05$.

Which of the following values cannot be probabilities of events?

$$\begin{array}{llllllll} \dfrac{1}{5} & 0.97 & -0.58 & 1.59 & \dfrac{5}{3} & 0.0 & -\dfrac{2}{7} & 1.0 \end{array}$$

Which of the following is not a characteristic of a continuous probability density function $f(x)$ ?

1. $f(x) \geq 0$ for all values of $x$
2. $f(x)$ is symmetric around the mean
3. the area under $f(x)$ over all values of $x$ equals one
4. all three are not

A distribution of scores has $\mu = 85$. The $z$-score for $X = 105$ is computed and a value of $z = -1.00$. Regardless of the value of the standard deviation, why must this $z$-score be incorrect?

At what level of significance can the null hypothesis be rejected?

If the hypothesis had been $H_0: \mu=34$ versus $H_1: \mu>34$, would the $P$-value have been larger or smaller?

The probability that event A will occur is P(A) = 0.64.

What is the probability (in decimal form) that event A will not occur? P(A) =

What are the odds for event A? ___ to ____

What are the odds against event A? ____ to _____

A police officer uses a motion detector to indicate whether a car is traveling faster than the speed limit (speeding). A speeding ticket will be issued to the driver of the car if the officer believes the driver is speeding, as indicated by the detector.

The situation is similar to using a null and an alternative hypothesis to decide whether to issue a ticket.

The hypotheses can be stated as follows:

• $H_0$: The driver is not speeding

• $H_a$: The driver is speeding.

Which of the following best describes the power of the test?

(A) The probability of issuing a ticket to a driver who is speeding

(B) The probability of issuing a ticket to a driver who is not speeding

(C) The probability of not issuing a ticket to a driver who is speeding

(D) The probability of not issuing a ticket to a driver who is not speeding

The viscosity of a chemical intermediate is measured every hour. Twenty samples each of size $n=1$ are in the following table.

$$\begin{array}{cccccc}\text { Sample } & \text { Viscosity } & \text { Sample } & \text { Viscosity } & \text { Sample } & \text { Viscosity } \\ 1 & 495 & 8 & 504 & 15 & 497 \\ 2 & 491 & 9 & 542 & 16 & 499 \\ 3 & 501 & 10 & 508 & 17 & 468 \\ 4 & 501 & 11 & 493 & 18 & 486 \\ 5 & 512 & 12 & 507 & 19 & 511 \\ 6 & 540 & 13 & 503 & 20 & 487 \\ 7 & 492 & 14 & 475 & & \end{array}$$

Using all the data, compute trial control limits for individual observations and moving-range charts. Determine whether the process is in statistical control. If not, assume that assignable causes can be found to eliminate these samples and revise the control limits.

Suppose G is a disconnected graph with N vertices, M edges and no circuits. (a) How many components does the graph have when N=9 and M=6? (b) How many components does the graph have when N=240 and M=236? Explain your answer.

The two-sample z-test can be viewed as a large sample test for the difference in means. Suppose that we need to compare the means of two independent Poisson distributions. Let $Y_{11}, Y_{12}, \ldots, Y_{1 n_{1}}$ be a random sample from a Poisson distribution with mean $\lambda_{\mathrm{I}} \text { and let } Y_{21}, Y_{22}, \ldots, Y_{2 n_{2}}$ and let be a random sample from a Poisson distribution with mean $\lambda_{2}.$ In applying the z-test we could take advantage of the fact that in the Poisson distribution both the mean and variance of the distribution are equal to $\lambda.$ Develop a variant of the z-test appropriate for this situation. Develop a large-sample CI for the difference in Poisson means.

An article in Experimental Brain Research ["Synapses in the Granule Cell Layer of the Rat Dentate Gyrus: Serial-Sectioning Study" ] showed the ratio between the numbers of symmetrical and total synapses on somata and azon initial segments of reconstructed granule cells in the dentate gyrus of a 12-week-old rat:

$$\begin{array}{llllll}0.65 & 0.90 & 0.78 & 0.94 & 0.40 & 0.94 \\ 0.91 & 0.86 & 0.53 & 0.84 & 0.42 & 0.50 \\ 0.50 & 0.68 & 1.00 & 0.57 & 1.00 & 1.00 \\ 0.84 & 0.9 & 0.91 & 0.92 & 0.96 & \\ 0.96 & 0.56 & 0.67 & 0.96 & 0.52 & \\ 0.89 & 0.60 & 0.54 & & & \end{array}$$

Find the $P$-value for the test.