Maths – a car company made a change to their fuel injection system

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A car company made a change to their fuel injection system that may improve gas mileage. To determine if this was the case, they selected 24 new cars at random from several days of production. The cars were randomly assigned to two groups of 12 cars each. One of these groups was fitted with the experimental fuel injection system, and the second was fitted with the current fuel injection system.

All the cars were driven around the same predetermined route, whose distance was known. By dividing the quantity of gas used by this driving distance, the automobile company was able to determine the gas mileage of each car. In summarizing the results, they found that the cars with the experimental fuel injection system had an average gas mileage of 33.4 mi/gal with a standard deviation of 1.3 mi/gal. Cars fitted with the current fuel injection system had an average gas mileage of 28.6 mi/gal with a standard deviation of 2.0 mi/gal.
How would you construct a test of hypothesis to determine whether or not cars with the experimental fuel injection system provide better gas mileage than cars with the current fuel injection system. (State the null and alternative hypotheses, give the test statistic that you would use, and state the most accurate distribution of this test statistic. Note, this is a one-sided test of hypothesis.) Because the improved system may result in the more consistent use of gas, don’t assume that the standard deviations of the two populations are equal.
Can you conclude at a significance level of 5% that the gas mileage of the experimental fuel injection system is better than that of the current fuel injection system?

 

Describe how you would conduct this test of hypothesis as a single-blind study. What are the advantages of designing the study in this fashion?

Given a commodity, the quantity of that commodity that a person selected at random will purchase will depend both on the Price “P” in dollars of the commodity and on Income “I” (in thousands of dollars) of the individual selected.

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Let “Q” be the number of shares of stock of a well-known company. Based on economic theory, the quantity “Q” that an individual will purchase this commodity is thought to be linearly related to both “I” and “P”. After collecting data, two regression analyses were performed to determine the actual relationship between “Q” and each of these independent variables. The following results were obtained:

Regression of Q as a function of Price:

  1. Based on the above analyses, express Q as linear function of Price and also as a linear function of Income. (Write down two linear relationships.)
  2. Based on the above tables, which of the two factors, Price or Income, will produce the most reliable estimate of Q?
  3. Based on the second model, how many more shares will a person selected at random purchase per each $1000 increase in his or her Income?
  4. Assuming the price per share of the stock is $54, how many shares would an individual selected at random either purchase or sell according to the first model above.

In an Oregon Lottery game called Cash Crossword, a player purchases a card for two dollars. Scratching the card reveals a crossword puzzle with a certain number of words that have been circled. Or, no words have been circled. The player wins, if there are one or more words that have been circled. The more words that have been circled, the more the player will win.

Assume that a total of 2,597,163 scratch-it cards are printed. Among these, the table below shows the number of winning cards printed for each possible number of circled words and the amount that the player can win for each of those winning cards. (All cards are distributed to outlets. No cards are printed with only one or two words circled.)

Each card can only win in a single category. That is, if a card contains ten circled words, the player can’t claim a win for 10 words as well as any smaller number of words.

a. Build a probability distribution table that gives the probability of winning for each different amount that can be won. (Be sure and include the probability that a player may win nothing.)
b. On average, what is the net amount that the player can expect to win or lose after purchasing a single card?
c. If on average, 37% of the scratch-it cards distributed are purchased, how much can the Oregon Lottery Commission expect to gain or lose from those sales? Do not consider administrative costs, like overhead, printing costs, distribution costs, etc.
d. How many non-winning scratch-it cards have been printed and distributed?

e. Assuming the same number of winning cards are printed, how many non-winning cards would need to be printed and distributed to make this game break-even for the lottery commission. (In this case, assume that all cards printed are sold. Do not consider administrative costs, like overhead, printing costs, distribution costs, etc.)

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