Problem 1: (10 points)

An experiment was conducted to examine the effect of age on heart rate when an individual is subjected to a specific amount of exercise. Ten male subjects were randomly selected from four age groups: 10-19, 20-39, 40-59, and 60-69. Each subject walked a treadmill at a fixed grade for a period of 12 minutes and the increases in heart rate (the difference before and after exercise) was recorded in beats per minute. The data is shown in the table below. Does this data provide sufficient evidence to indicate a difference in mean increase in heart rate among the four age groups? Test using α = .05.

1. State the null and alternative hypothesis ( 2 points ).
2. Develop and attach an ANOVA table. Please identify the F-statistic and the p-value. ( 4 points ).
3. What is your decision regarding the null hypothesis ( 2 points ).
4. Explain "Does this data provide sufficient evidence to indicate a difference in mean increase in heart rate among the four age groups? " ( 2 points )

P roblem 2: ( 10 Points )

The space shuttle Challenger was in launch position the morning of January 28, 1986, with seven astronauts and passengers on board. Just prior to the launch, sheets of ice clung to the fuselage. Moments later, with national television coverage as it blasted into orbit, the shuttle disintegrated in a catastrophic explosion. The remains of the astronauts and passengers were never recovered.

Thiokol Corporation manufactures the two solid-fuel rocket motors that propelled the shuttle into space. The night before the catastrophe, executives of Thiokol and NASA debated whether they should launch the shuttle according to schedule or postpone the mission. The weather report called for a temperature of 31 F at blast off. At the conclusion of the debate, Thiokol executives recommended that the shuttle be launched on schedule because they felt that they did not have conclusive evidence that the low temperature would influence the capability of the solid-fuel rocket motors.

From April 12, 1981 to January 12, 1986, the space shuttle had flown 24 successful missions. Six primary O-rings were used to seal the sections of the two solid-fuel rocket motors. On several flights, the motors had experienced O-ring erosion or gas blow-by incidents. O-ring incidents are extremely dangerous because a failed O-ring can allow super-hot gases to escape the motors and ignite the liquid hydrogen fuel tank. On one of the 24 flights, the motors were not recovered. The number of erosion or blow-by incidents and the temperature at launch for 23 successful flights are given in the following table.

1. Create a scatterplot of the data and include the regression equation. The number of O-ring failures is the dependent variable, Y, and the independent variable is temperature, X. (4 points)
2. Is there a significant relationship between number of failures and temperature? Please explain how you have made this decision. You may use α = . 05. (2 points)
3. It is important to determine whether colder temperatures have a higher number of O-ring failures. Suppose you are trying to convince Thiokol and NASA that low temperature at launch time is a danger. Set up and perform an appropriate hypothesis test. Make sure you state the null and alternative, the p-value, and your conclusions. You may use α = . 05
4. Predict the number of 0-ring failures for a launch at 31 F, the temperature when Challenger was launched. Does it look as if there was a serious problem? Please explain.

Problem 3: (10 points)

It has been suggested that treatment of a plastic used in optic lenses will improve wear. Four different treatments are to be tested. To determine whether any differences in mean wear exist among treatments, 28 castings were made from a single formulation of the plastic and seven castings were randomly assigned to each of the treatment groups. "Wear" was determined by measuring the increase in "haze" after 200 cycles of abrasion (better wear is indicated by a small increase).

1. Using SPSS to complete an analysis of your choice, can you determine a significant difference in mean wear among the four treatments? Use α = .05. ( 5 points )
2. Estimate the mean difference in haze increase between treatments B and C using a 99% confidence interval ( 3 points ).
3. Find a 90% confidence interval for the mean wear for lenses receiving treatment A ( 2 points ).

Problem 4: (10 points)

Suppose a random sample of twelve families had the following annual income and savings. Using SPSS, answer the following questions.

1. What is the regression equation for Savings (S) ( 5 points ).
2. Graph the points and a fitted line (as well as the slope). ( 3 points )
3. If income is 25, what can you estimate savings to be? ( 2 points )

Problem 6 : (15 points)

The MacBurger restaurant chain claims that the mean waiting time of customers is 3 minutes with a population standard deviation of 1 minute. The quality-assurance department found in a sample of 50 customers at the Warren Road MacBurger that the mean waiting time was 3.05 minutes. At the .05 significance level, can we conclude that the mean waiting time is less than 3 minutes?

1. Is this a one-tailed or two-tailed test? Please draw and label the normal curve. ( 6 points )
2. Please state in words and symbols the null and alternative hypotheses. ( 4 points )
3. Please state what method or technique you used to analyze the data to answer the question of whether or not the mean waiting time is less than 3 minutes. ( 3 points )
4. Can you reject the null hypothesis? ( 2 points )

Problem 7: (10 points)

Citrus Clean is a new all-purpose cleaner being test-marketed by placing displays in three different locations within various supermarkets. The number of 12-ounce bottles sold from each location within the super market is report below.

At the .05 significance level, is there a difference in the mean number of bottles sold at the three locations?

1. State the null and alternative hypothesis. ( 2 points )
2. Develop and ANOVA table. ( 4 points )
3. What is your decision regarding the null hypothesis? ( 2 points )
4. Explain how you made your decision regarding your answer to part c. ( 2 points )
   Problem 8: (20 points)
   Salsberry Realty sells homes along the east coast of the U.S. One of the questions most frequently asked by prospective buyers is: If we purchase this home, how much can we expect to pay to heat it during the winter? The research department at Salsbury has been asked to develop some guidelines regarding heating costs for single-family homes. Three variables are thought to relate to the heating costs:
   1. the mean daily outside temperature,
   2. the number of inches of insulation in the attic, and
   3. the age, in years, of the furnace.

To investigage, Salsbury’s research department selected a random sample of 20 recently sold homes. It determined the cost to heat each home last January, as well as the January outside temperature in the region, the number of inches of insulation in the attic, and the age of the furnace. The sample information is reported on the next page:

1. What is the dependent variable? What are the independent variables? ( 2 points ).
2. Using SPSS to analyze the data and the corresponding output, what is the regression equation? ( 2 points )
3. If the mean outside temperature for the month of January is 30 degrees, there are 5 inches of insulation in the attic, and the furnace is 10 years old, what is the estimated heating cost for January? ( 2 points ).
4. What is the regression coefficient for mean outside temperature? What relationship does it have with heating cost? Explain what happens to heating cost as the outside temperature increases. ( 3 points )
5. If we increase temperature by 1 degree and hold the other variables constant, what can we expect will happen to the heating cost? ( 1 points )
6. If the mean temperature in Boston is 25 degrees and it is 35 degrees in Philadelphia, all other things being the same, what can we expect the difference to be in heating cost? ( 2 points )
7. What is the regression coefficient for attic insulation? What relationship does it have with heating cost? ( 4 points )
8. What is the regression coefficient for age of furnace? What relationship does it have with heating cost? ( 4 points )

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