Chapter 17

A city is divided into 4 voting precincts, \(A, B, C,\) and \(D\). Table 17.20 shows the results of mayoral election held for two candidates, a Republican and a Democrat.$$ \begin{array}{c|c|c|c} \hline \text { Precinct } & \text { Number voters } & \text { Republican } & \text { Democrat } \\ \hline \mathrm{A} & 10,000 & 4,200 & 5,800 \\ \mathrm{~B} & 15,000 & 7,100 & 7,900 \\ \mathrm{C} & 17,000 & 8,200 & 8,800 \\ \mathrm{D} & 18,000 & 12,400 & 5,600 \\ \hline \end{array}$$ Assuming random selection, what is the probability, given as a percentage, that a voter: (a) Lives in precinct \(B ?\) (b) Is a Republican? (c) \(\operatorname{Both}(\) a) and \((\mathrm{b})\) (d) Is Republican given that he or she lives in precinct \(B ?\) (e) Lives in precinct \(B\) given that he or she is Republican?

In 10 packages the number of M\&M's was $$ 56,53,54,54,52,55,52,53,55,55 $$ (a) What is the mean number of M\&M's per package? (b) Is the mean a good description of the count for a package of M\&M's? Explain your reasoning.

For his term project in biology, Robert believed he could increase the weight of mice by feeding them a hormone. Do his results, in Table 17.21 , support the claim that the hormone increases weight? $$ \begin{array}{c|c|c|c} \hline & \text { Weight increase } & \text { No weight increase } & \text { Total } \\ \hline \text { Fed hormone } & 120 & 30 & 150 \\ \hline \text { Not fed hormone } & 25 & 25 & 50 \\ \hline \text { Total } & 145 & 55 & 200 \\ \hline \end{array}$$

Suppose you record the hours of daylight each day for a year in Tucson, Arizona, and find the mean. (a) What do you expect for an approximate mean? (b) How would your data compare with a student doing the same project in Anchorage, Alaska? (c) How would your mean compare with a student doing the same project in Anchorage, Alaska?

Face recognition systems pick faces out of crowds at airports to see if any matches occur with law enforcement databases. Performance of the systems can be affected by lighting, gender and age of the target, and age of the database. In Problems \(24-27,\) the tables give identification rates for faces under various conditions. Decide whether the rate of recognition is independent of the given factor. Table 17.22 compares face recognition under different lighting conditions.$$ \begin{array}{c|c|c} \hline \text { Lighting } & \text { Did recognize } & \text { Did not recognize } \\ \hline \text { Indoors } & 900 & 100 \\ \hline \text { Outdoors } & 300 & 300 \\ \hline \end{array} $$

Face recognition systems pick faces out of crowds at airports to see if any matches occur with law enforcement databases. Performance of the systems can be affected by lighting, gender and age of the target, and age of the database. In Problems \(24-27,\) the tables give identification rates for faces under various conditions. Decide whether the rate of recognition is independent of the given factor. Table 17.23 compares face recognition for men and women.$$ \begin{array}{c|c|c} \hline \text { Gender } & \text { Did recognize } & \text { Did not recognize } \\ \hline \text { Men } & 78 & 22 \\ \hline \text { Women } & 117 & 33 \\ \hline \end{array} $$

Find the mean of each data set: (a) Five readings equaling (not totaling) \(120,\) three readings equaling 130 , two readings equaling 140 , four readings equaling 150 , and one reading equaling 160 . (b) Three readings equaling \(x_{1}\), six readings equaling \(x_{2}\), seven readings equaling \(x_{3}\), five readings equaling \(x_{4},\) and four readings equaling \(x_{5}\). (c) \(n_{1}\) readings equaling \(x_{1}, n_{2}\) readings equaling \(x_{2}\), and so on, up to \(n_{5}\) readings equaling \(x_{5}\).

Face recognition systems pick faces out of crowds at airports to see if any matches occur with law enforcement databases. Performance of the systems can be affected by lighting, gender and age of the target, and age of the database. In Problems \(24-27,\) the tables give identification rates for faces under various conditions. Decide whether the rate of recognition is independent of the given factor. Table 17.24 compares face recognition for different ages.$$ \begin{array}{c|c|c} \hline \text { Age } & \text { Did recognize } & \text { Did not recognize } \\\ \hline 18-22 & 248 & 152 \\ \hline 38-42 & 222 & 78 \\ \hline \end{array} $$

Suppose two samples of 5 values are taken from a population $$ a: 8,9,4,7,5 \text { and } b: 6,9,10,5,7 $$ (a) Find \(\bar{a}\) and \(\bar{b}\). (b) Find the mean of the sample you get by combining the two samples. (c) Is the mean of the combined sample equal to the mean of the two values \(\bar{a}\) and \(\bar{b}\) ? (d) Explain your answer in (c) algebraically.

Face recognition systems pick faces out of crowds at airports to see if any matches occur with law enforcement databases. Performance of the systems can be affected by lighting, gender and age of the target, and age of the database. In Problems \(24-27,\) the tables give identification rates for faces under various conditions. Decide whether the rate of recognition is independent of the given factor. Table 17.25 compares face recognition when using a fresh, same-day database image to face recognition when using an older database image.$$ \begin{array}{c|c|c|} \hline \text { Image } & \text { Did recognize } & \text { Did not recognize } \\\ \hline \text { Fresh } & 47 & 3 \\ \hline \text { Older } & 68 & 12 \\ \hline \end{array} $$

Suppose two samples of values are taken from a population $$ a: 8,2,4,7,5 \text { and } b: 6,9,10,5,7,8,2,5 $$ (a) Find \(\bar{a}\) and \(\bar{b}\). (b) Find the mean of the sample you get by combining the two samples. (c) Is the mean of the combined sample equal to the mean of the two values \(\bar{a}\) and \(\bar{b}\) ? (d) Explain why the means in (c) are the same or different.

A high-tech company makes silicon wafers for computer chips, and tests them for defects. The test identifies \(90 \%\) of all defective wafers, and misses the remaining \(10 \% .\) In addition, it misidentifies \(20 \%\) of all non- defective wafers as being defective. (a) Suppose 5000 wafers are made. Of the \(5 \%\) of these wafers that contain defects, how many are correctly identified by the test as being defective? (b) How many of the non-defective wafers are incorrectly identified by the test as being defective? (c) What is the probability, given as a percentage, that a wafer identified as defective is actually defective?

A researcher has a colony of ants with a mean weight of \(\mu,\) and takes a sample of 10 ants with weights \(w_{1}\) \(\ldots, w_{10} .\) Decide whether each of the following is the mean of the sample. Explain. (a) \(\sum_{\bar{w}} w_{i}\) (c) \(\frac{1}{10} \sum_{i=1}^{10} w_{i}\) (d) \(\mu\) (e) \(\left(w_{1}+w_{2}+\cdots+w_{10}\right) / 10\) (f) \(\frac{\mu}{10}\)