Suppose that you have n values to put into an empty binary search tree.

1. In how many different orders can you add the n values to the tree? This is not the same as the number of possible binary search trees for n values. Explain why.
2. What is the probability that a randomly constructed binary search tree has worst-case performance? Hint: Compute the fraction of the total number of possible orders that results in the worst case.

The following data on x = score on a measure of test anxiety and y = exam score for a sample of n = 9 students are consistent with summary quantities given in the paper "Effects of Humor on Test Anxiety and Performance" (Psychological Reports 11999]: 1203-1212): Higher values for x indicate higher levels of anxiety. a. Construct a scatterplot, and comment on the features of the plot. b. Does there appear to be a linear relationship between the two variables? How would you characterize the relationship? c. Compute the value of the correlation coefficient. Is the value of r consistent with your answer to Part (b)? d. Is it reasonable to conclude that test anxiety caused poor exam performance? Explain.

$$\begin{matrix} \text{x} & \text{23} & \text{14} & \text{14} & \text{0} & \text{17} & \text{20} & \text{20} & \text{15} & \text{21}\\ \text{y} & \text{43} & \text{59} & \text{48} & \text{77} & \text{50} & \text{52} & \text{46} & \text{51} & \text{51}\\ \end{matrix}$$

A one-sided confidence interval for p can be expressed as $p<\hat{p}+E \text { or } p>\hat{p}-E$, where the margin of error E is modified by replacing $z_{\alpha / 2}$ with $z_{\alpha}$. If Air America wants to report an on-time performance of at least x percent with 95% confidence, construct the appropriate one-sided confidence interval and then find the percent in question. Assume that a simple random sample of 750 flights results in 630 that are on time.

In the 2008 baseball season, first baseman Ryan Howard of the Philadelphia Phillies led the Major Leagues with 48 home runs (HR) and 146 runs batted in (RBI). Overall, using all players with at least 300 plate appearances, the mean number of HR was $14.9$ with a standard deviation of $9.8$, whereas the mean number of RBI was $62.5$ with a standard deviation of $25.5$. Which was the more impressive PERFORMANCE? Explain, using the methods of this chapter.

One yardstick for measuring how steadily if slowly athletic performance has improved is the mile run. In 1954, Roger Bannister of Britain cracked the 4-minute mark, setting the record for running a mile in 3 minutes, 59.4 seconds, or 239.4 seconds. In the half-century since then, the record has decreased by 0.3 second per year. Express the record time for the mile run, M, as a function of the number of years after 1954, x.

A new surgical procedure is said to be successful 80% of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another. a. What is the probability that all five operations are successful? b. What is the probability that exactly four are successful? c. What is the probability that less than two are successful? d. If less than two operations were successful, how would you feel about the performance of the surgical team?

An office building requires a heat transfer rate of $20\ \mathrm{kW}$ to maintain the inside temperature at $21^{\circ} \mathrm{C}$ when the outside temperature is $0^{\circ} \mathrm{C}$. A vapor-compression heat pump with Refrigerant $22\mathrm{a}$ as the working fluid is to be used to provide the necessary heating. The compressor operates adiabatically with an isentropic efficiency of $82 \%$. Specify appropriate evaporator and condenser pressures of a cycle for this purpose assuming $\Delta T_{\text {cond }}=\Delta T_{\text {cvap }}=10^{\circ} \mathrm{C}$, as shown in figure. The states are numbered as in figure. The refrigerant exits the evaporator as saturated vapor and exits the condenser as saturated liquid at the respective pressures. Determine the (a) mass flow rate of refrigerant, in $\mathrm{kg} / \mathrm{s}$. (b) compressor power, in $\mathrm{kW}$. (c) coefficient of performance and compare with the coefficient of performance for a Carnot heat pump cycle operating between reservoirs at the inside and outside temperatures, respectively.. Compare the results with those of previous problem and discuss.

A windows air conditioner that consumes of 1 kW of electricity when running and has a coefficient of performance of 3 is placed in the middle of a room and is plugged in. The rate of cooling or heating this air conditioner will provide to the air in the room when running is (a) 3 kJ/s, cooling (b) 1 kJ/s, cooling (c) 0.33 kJ/s, heating (d) 1 kJ/s, heating (e) 3 kJ/s, heating

Cell Design produces cell phone covers for all makes and models of cell phones. Cell Design sells 1,050,000 units each year at a price of $10 per unit and a contribution margin of 40%.

A survey of Cell Design customers over the past 12 months indicates that customers were very satisfied with the products but a disturbing number of customers were disappointed because the products they purchased did not fit their phones. They then had to hassle with returns and replacements.

Cell Design’s managers want to modify their production processes to develop products that more closely match Cell Design’s specifications because the quality control in place to prevent ill-fitting products from reaching customers is not working very well.

The current costs of quality are as follows:

$$\begin{array}{lr} \text{Prevention costs}& \$210,000\\ \text{Appraisal costs}& \$100,000\\ \text{Internal failure costs}&\\ \quad\text{Rework}& \$420,000\\ \quad\text{Scrap}& \$ 21,000\\ \text{External failure costs}&\\ \quad\text{Product replacements}& \$315,000\\ \quad\text{Lost sales from customer returns}& \$787,500\\ \text{The QC manager and controller have forecast the following}& \\ \quad\text{additional costs to modify the production process.}&\\ \text{CAD design improvement}& \$150,000\\ \text{Improve machine calibration to specifications}& \$137,500\\ \end{array}$$

1. Which cost of quality category are managers focusing on? Why?

2. If the improvements result in a 60% decrease in customer replacement cost and a 70% decrease in customer returns, what is the impact on the overall COQ and the company’s operating income? What should Cell Design do? Explain.

3. Calculate prevention, appraisal, internal failure, and external failure costs as a percentage of total quality costs and as a percentage of sales before and after the change in the production process. Comment briefly on your results.

In 2012, the New York Yankees won 95 games and spent $\$ 198$ million on salaries for their players (USA Today). Is there a relationship between salary and team performance in Major League Baseball? For the 2012 season, a linear model fit to the number of Wins (out of 162 regular season games) from the team Salary $(\$ M)$ for the 30 teams in the league is:

$$\widehat{\text { Wins }}=76.45+0.046 \text { Salary. }$$

c) What does the slope mean in this context?

Ms. Ledbetter wants to determine if the new review activity she developed will improve student performance on unit exams. She randomly separates 160 students into two groups. Group A reviews for the unit exam in the traditional manner they have always used. Group B participates in the new review activity. After reviewing, both groups are given the same unit exam and their scores are compared. Identify the independent and dependent variables for this experiment.

Alvarez Manufacturing Inc. is a job shop. The management of Alvarez Manufacturing Inc. uses the cost information from the job sheets to assess cost performance. Information on the total cost, product type, and quantity of items produced is as follows:

$$\begin{array}{lclcr} \text { Date } & \text { Job No. } & \text { Product } & \text { Quantity } & \text { Amount } \\ \hline \text { Jan. 2 } & 1 & \text { TT } & 520 & \$ 16,120 \\ \text { Jan. 15 } & 22 & \text { SS } & 1,610 & 20,125 \\ \text { Feb. 3 } & 30 & \text { SS } & 1,420 & 25,560 \\ \text { Mar. 7 } & 41 & \text { TT } & 670 & 15,075 \\ \text { Mar. 24 } & 49 & \text { SLK } & 2,210 & 22,100 \\ \text { May 19 } & 58 & \text { SLK } & 2,550 & 31,875 \\ \text { June 12 } & 65 & \text { TT } & 620 & 10,540 \\ \text { Aug. 18 } & 78 & \text { SLK } & 3,110 & 48,205 \\ \text { Sept. 2 } & 82 & \text { SS } & 1,210 & 16,940 \\ \text { Nov. 14 } & 92 & \text { TT } & 750 & 8,250 \\ \text { Dec. 12 } & 98 & \text { SLK } & 2,700 & 52,650 \end{array}$$

a. Develop a graph for each product (three graphs) with Job Number (in date order) on the horizontal axis and Unit Cost on the vertical axis. Use this information to determine Alvarez Manufacturing Inc.’s cost performance over time for the three products.

b. What additional information would you require in order to investigate Alvarez Manufacturing Inc.’s cost performance more precisely?

Of the 281 players with at least 300 plate appearances in the 2008 Major League Baseball season, Jeff Mathis of the Los Angeles Angels had the lowest batting average ( $0.194)$ and Brad Wilkerson of the Toronto Blue Jays scored the fewest runs (21). Overall, the mean batting average was $0.272$, with a standard deviation of $0.027$, and the mean number of runs scored was $65.0$, with a standard deviation of 23.6. Which PERFORMANCE was worse? Explain, using the methods of this chapter.

Katie and Kim are attending a theater performance. Katie's seat is at floor level. She looks down at an angle of 18$^{\circ}$ to see the orchestra pit. Kim's seat is in the balcony directly above Katie. Kim looks down at an angle of 42$^{\circ}$ to see the pit. The horizontal distance from Katie's seat to the pit is 46ft. What is the vertical distance between Katie's seat and Kim's seat? Round to the nearest inch.