A car manufacturer is trying to develop a new sports car. Engineers are hoping that the average amount of time that the car takes to go from 0 to 60 miles per hour is below 6 seconds. The manufacturer tested 12 of the cars and clocked their performance times. Three of the cars clocked in at $5.8$ seconds, 5 cars at $5.9$ seconds, 3 cars at $6.0$ seconds, and 1 car at $6.1$ seconds. At the $5 \%$ level of significance, test if the new sports car is meeting its goal to go from 0 to 60 miles per hour in less than 6 seconds. Assume a normal distribution for the analysis.

Construct the null and the alternative hypotheses for the following claims:

a. "I am going to get the majority of the votes to win this election."

b. "I suspect that your 10 -inch pizzas are, on average, less than 10 inches in size."

c. "I will have to fine the company since its tablets do not contain an average of $250 \mathrm{mg}$ of ibuprofen as advertised."

An economist reports that 560 out of a sample of 1,200 middle-income American households actively participate in the stock market.

a. Construct the 90% confidence interval for the proportion of middle-income Americans who actively participate in the stock market.

b. Can we conclude that the proportion of middleincome Americans who actively participate in the stock market is not 50% ?

TGH are a public limited company which makes sports shoes (trainers). It makes a variety of designs for the mass market. The mass market is for young people who want fashionable sports shoes as well as those who play sport. The Directors recently decided to target a niche market which is for a particular market segment of customers who have a medical problem with their feet and need specially designed shoes.

$$\begin{matrix} \text{a) What is meant by a 'niche market'? } & \text{[2]}\\ \text{b) Identify two examples of the different markets for TGH products. } & \text{[2]}\\ \text{c) Identify and explain two reasons why TGH sell to a niche market. } & \text{[4]}\\ \text{d) Identify and explain two advantages to TGH of segmenting the market for sports shoes.} & \text{[6]}\\ \text{e) Do you think the Directors of TGH were right to target a niche market or should they have stayed with just a mass market? Justify your answer. } & \text{[6]}\\ \end{matrix}$$

According to a recent survey, high school girls average 100 text messages daily (The Boston Globe, April 21, 2010). Assume the population standard deviation is 20 text messages. Suppose a random sample of 50 high school girls is taken.

a. What is the probability that the sample mean is more than 105 ?

b. What is the probability that the sample mean is less than 95 ?

c. What is the probability that the sample mean is between $95$ and $105 ?$

Customers at Costco spend an average of $\$ 130$ per trip (The Wall Street Journal, October 6,2010 ). One of Costco's rivals would like to determine whether its customers spend more per trip. A survey of the receipts of 25 customers found that the sample mean was $\$ 135.25$. Assume that the population standard deviation is $\$ 10.50$ and that spending follows a normal distribution.

b. Calculate the value of the test statistic and the p-value.

D & F Limited own a number of farms which grow fruit. The fruit is sold to large supermarkets abroad. They also grow coffee which is sold to companies that process and brand the coffee which is also then sold to large supermarkets abroad. The world price of coffee has recently dropped by 20 per cent. There has been a growing demand for fruit from developing countries with rising incomes as well as an increasing demand from developed countries which have seen a growing trend towards 'healthy eating', encouraged by government policies to tackle their obesity problem.

$$\begin{matrix} \text{a) What customer needs are being satisfied by D \& F Limited? } & \text{[2]}\\ \text{b) Identify two reasons why the price of coffee might be falling. } & \text{[2]}\\ \text{c) Identify and explain two reasons why supermarkets are demanding more fruit.} & \text{[4]}\\ \text{d) Identify and explain two ways the Directors of D\& F Limited could respond to increased competition from farms which now grow fruit instead of vegetables. } & \text{[6]}\\ \text{e) Do you think D \& F Limited should change production from coffee to growing more fruit? Justify your answer.} & \text{[6]}\\ \end{matrix}$$

A local brewery wishes to ensure that an average of 12 ounces of beer is used to fill each bottle. In order to analyze the accuracy of the bottling process, the bottler takes a random sample of 48 bottles. The sample mean weight and the sample standard deviation of the bottles are $11.80$ ounces and $0.8$ ounce, respectively.

a. State the null and the alternative hypotheses to test if the accuracy of the bottling process is compromised.

b. Do you need to make any assumption regarding the population for testing?

c. Calculate the value of the test statistic and the $p$-value.

d. At $\alpha=0.05$, what is the conclusion to the test? Make a recommendation to the bottler. Based on the average predictions of 47 members of the National Association of Business

In September 2007, U.S. home prices fell at a record pace, and price declines in Los Angeles and Orange counties in California outpaced other major metropolitan areas (Los Angeles Times, November 28, 2007). The report was based on the Standard & Poor's/Case-Shiller index that measures the value of single-family homes based on their sales histories. According to this index, the prices in San Diego dropped by an average of $9.6 \%$ from a year earlier. Assume that the survey was based on recent sales of 34 houses in San Diego that also resulted in a standard deviation of $5.2 \%$. Can we conclude that the mean drop of all home prices in San Diego is greater than the $7 \%$ drop in Los Page 344 Angeles? Use a $1 \%$ level of significance for the analysis.

Economists (NABE), the U.S. gross domestic product (GDP) will expand by $3.2 \%$ in 2011 (The Wall Street Journal, May 23, 2010). Suppose the sample standard deviation of their predictions was $1 \%$. At the $5 \%$ significance level, test if the mean forecast GDP of all NABE members is greater than $3 \%$.

Small, energy-efficient, Internet-centric, new computers are increasingly gaining popularity (The New York Times, July 20, 2008). Some of the biggest companies are wary of the new breed of computers because their low price could threaten PC makers' already thin profit margins. An analyst comments that the larger companies have a cause for concern since the mean price of these small computers has fallen below $\$ 350$. She examines six popular brands of these small computers and records their retail prices as:

$$\begin{array}{|l|l|l|l|l|l|} \hline 322 & 269 & 373 & 412 & 299 & 389 \\ \hline \end{array}$$

d. At the $5 \%$ significance level, what is the conclusion to the test? Should the larger computer companies be concerned?

Small, energy-efficient, Internet-centric, new computers are increasingly gaining popularity (The New York Times, July 20, 2008). Some of the biggest companies are wary of the new breed of computers because their low price could threaten PC makers' already thin profit margins. An analyst comments that the larger companies have a cause for concern since the mean price of these small computers has fallen below $\$ 350$. She examines six popular brands of these small computers and records their retail prices as:

$$\begin{array}{|l|l|l|l|l|l|} \hline 322 & 269 & 373 & 412 & 299 & 389 \\ \hline \end{array}$$

d. At the $5 \%$ significance level, what is the conclusion to the test? Should the larger computer companies be concerned?

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5 kg/m.

How much work is required to wind the entire chain onto the cylinder using the winch?

A heavy-duty shock absorber is compressed 2 cm from its equilibrium position by a mass of 500 kg. How much work is required to compress the shock absorber 4 cm from its equilibrium position? 1A mass of 500 kg exerts a force (in newtons) of 500 g, where $g \approx 9.8 \mathrm { m } / \mathrm { s } ^ { 2 }$.)