CHAPTER 7 REVIEW



CHAPTER 7 REVIEW

The following questions are in a True/False format. The answers to these questions will frequently depend on remembering facts, understanding of the concepts, and knowing the statistical vocabulary. Before answering these questions, be sure to read them carefully!

33) A random variable is discrete if the value of the random variable depends upon the outcome of a chance experiment.

ANS: False

34) A random variable is continuous if the set of possible values includes an entire interval on the number line.

ANS: True

35) For a discrete random variable x, [pic].

ANS: True

36) For a discrete random variable, [pic].

ANS: False

37) Using the notation C = continuous and D = discrete, indicate whether each of the random variables are discrete or continuous.

|a) |The number of defective lights in your school's main hallway |

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|b) |The barometric pressure at midnight |

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|c) |The number of staples left in a stapler |

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|d) |The number of sentences in a short story |

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|e) |The average oven temperature during the cooking of a turkey |

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|f) |The number of lightning strikes during a thunderstorm |

ANS: a) D b) C c) D d) D e) C f) D

38) The famous physicist, Ernest Rutherford, was a pioneer in the study of radioactivity using electricity. In one experiment he observed the number of particles reaching a counter during time 1700 intervals of 7.5 seconds each. The number of intervals that had 0 - 4 particles reaching the counter is given in the table below.

| |Number of time |

| |intervals with k |

|k particles |particles |

|0 |57 |

|1 |203 |

|2 |383 |

|3 |525 |

|4 |532 |

Let the random variable x = number of particles counted in a 7.5 second time period.

|a) |Fill in the table below with the estimated probability distribution of x, and sketch a probability histogram for x. |

| | |

| |Probability distribution | |

| | | | |

| |x |P(x) | |

| |0 | | |

| |1 | | |

| |2 | | |

| |3 | | |

| |4 | | |

| |Probability histogram |

| |[pic] |

|b) |Using the estimated probabilities in part (a), estimate the following: |

| | |

| |i) |P(x = 1), the probability that 1 particle was counted in 7.5 seconds. |

| | | |

| |ii) |P(x < 3), the probability that fewer than 3 particles were counted. |

| | | |

| |iii) |P(x ≥ 3), the probability that at least 3 particles were counted. |

ANS:

|a) |Probability Distribution: |

| | |

| |Histogram: Number of particles |

| |[pic] |

| | |

|b) |i) |P(x = 1) = 0.1194 |

| |ii) |P(x < 3) = 0.3782 |

| |iii) |P(x ≥ 3) = 0.6218 |

39) The sales of a newsmagazine vary from week to week, depending on the importance of the news of the previous week. From empirical data over many years, the following distribution of sales (rounded to the nearest million) has been carefully constructed. What are the mean and standard deviation of this distribution?

|X |0 |1 |2 |3 |

|P(X) |0.15 |0.25 |0.40 |0.20 |

ANS:

|μx |[pic] |

|σx |[pic] |

| |= (−1.65)2(0.15) + (−0.65)2(0.25) + (0.35)2(0.40) + (1.35)2(0.20) |

| |[pic] |

| |= 0.9631 |

40) Determine the following areas under the standard normal (z) curve.

|a) |The area under the z curve to the left of 2.53 |

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|b) |The area under the z curve to the left of −1.33 |

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|c) |The area under the z curve to the right of 0.76 |

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|d) |The area under the z curve to the right of −1.47 |

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|e) |The area under the z curve between −1 and 3 |

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|f) |The area under the z curve between −2.6 and −1.2 |

ANS:

|a) |0.9943 |

|b) |0.0918 |

|c) |0.2236 |

|d) |0.9292 |

|e) |0.8400 |

|f) |0.1104 |

41) A gasoline tank for a certain model car is designed to hold 12 gallons of gas. Suppose that the actual capacity of the gas tank in cars of this type is well approximated by a normal distribution with mean 12.0 gallons and standard deviation 0.2 gallons. What is the probability that a randomly selected car of this model will have a gas tank that holds at most 11.7 gallons?

ANS: [pic]

42) The owners of the Burger Emporium are looking for new supplier of onions for their famous hamburgers. It is important that the onion slice be roughly the same diameter as the hamburger patty. After careful analysis, they determine that they can only use onions with diameters between 9 and 10 cm. Company A provides onions with diameters that are approximately normally distributed with mean 10.3 cm and standard deviation of 1.2 cm. Company B provides onions with diameters that are approximately normally distributed with mean 10.6 cm and standard deviation of 0.9 cm. Which company provides the higher proportion of usable onions? Justify your choice with an appropriate statistical argument.

ANS: The onions from Company A will be between 9 and 10 cm diameter with probability [pic].

The onions from Company B will be between 9 and 10 cm diameter with probability [pic].

There are a greater proportion of the onions from Company A that are the correct size.

43) The time that it takes a randomly selected employee to perform a certain task is approximately normally distributed with a mean value of 120 seconds and a standard deviation of 20 seconds. The slowest 10% (that is, the 10% with the longest times) are to be given remedial training. What times qualify for the remedial training?

ANS: To qualify for remedial training, the time would be greater than or equal to the highest 10% of the times. [pic]. Solve [pic] to get x > 145.6.

44) While playing Monopoly, Andi estimated the probabilities of the non-zero rents according to the following probability distribution:

|x |$2 |$14 |$20 |$100 |

|P(x) |0.40 |0.20 |0.20 |0.20 |

Consider the random variable x = dollar amount in rent collected in a Monopoly roll.

|a) |If the table above specifies the probability distribution of x, what is the mean of the random variable x? |

| | |

|b) |If the probabilities are associated with the outcomes as in the table above, what is the standard deviation of the |

| |random variable x? |

ANS:

|a) |[pic] |

| | |

|b) |[pic] |

45) At the last home football game, the senior football players walk through a specially constructed welcoming arch, 2 abreast. It is considered unseemly to bump each other on the way through, so the arch must be wide enough for two players to go through. The distribution of widths of football players with shoulder pads on is approximately normal; the mean padded football player width is 30 inches, and the standard deviation is 5 inches. Let random variable w = width in inches of a randomly selected padded football player.

|a) |Since this is a statistics test, the carpenter who will be constructing the arch has only metric measuring tools, and |

| |must convert all the information above to metric measures. 1 inch = 2.54 centimeters, and we define random variable m = |

| |width in centimeters of a randomly selected padded football player. What are the mean and standard deviation of m? |

| | |

|b) |Suppose the football players are paired randomly to go through the arch. Define random variable v = w1 + w2 to be the |

| |collective width (in inches) of two randomly selected football players. What are the mean and standard deviation of v? |

| | |

|c) |Suppose that in the original specifications of the arch a 10 inch separation of the football players is specified so |

| |that the football players have room to avoid each other. This leads to random variable a = 10 + w1 + w2. Describe how |

| |this addition of 10 inches of "wiggle room" would change the mean and standard deviation in part (b). Do not recalculate|

| |the mean and standard deviation. |

ANS:

|a) |[pic] |

| |[pic] |

| | |

|b) |[pic] |

| |[pic] |

| | |

|c) |The mean would increase by 10". [to 70"] |

| |The standard deviation would remain unchanged. [7.07"] |

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