14. In the New York State Win 4 lottery, you place a bet by selecting four digits. Repetition is allowed, and the winning requires that your sequence of four digits matches the four digits that are later drawn. Assume that you place one bet with a sequence of four digits.

a. Use the multiplication rule to find the probability that your first two digits match those drawn and your last two digits do not match those drawn. That is, find P(MMXX), where M denotes a match and X denotes a digit that does not match the winning number.

b. Beginning with MMXX, make a complete list of the different possible arrangements of two matching digits and two digits that do not match, then find the probability for each entry in the list.

c. Based on the preceding results, what is the probability of getting exactly two matching digits when you select four digits for the Win 4 lottery game?

Chapter 5-4 Exercise 10

When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 580 peas, and Mendel theorized that 25% of them would be yellow peas.

a. If Mendel’s theory is correct, find the man and standard deviation for the numbers of yellow peas in such groups of 580 offspring peas.

b. The actual results consisted of 152 yellow peas. Is that result unusually high? What does this result suggest about Mendel’s theory?

16. The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter r is used at a rate of 6%.

a. Find the mean and standard deviation for the number of times the letter r will be found on a typical page of 2600 characters.

b. In an intercepted ciphered message sent to Iran, a page of 2600 characters is found to have the letter r occurring 178 times. Is this unusually low or high?

Chapter 6-2 Exercise 46

Draw a graph and find the probability of the given scores using table A-2 on page 249 or technology rounded to four decimal places:

About ________% of the area is between z = -2 and z = 2 (or within two standard deviations of the mean).

Chapter 6-3 Exercise 16

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Draw a graph in each case.)

16. Find the probability that a randomly selected adult has an IQ between 110 and 120 (referred to as a bright normal).

32. After 1964, quarters were manufactured so that he weights had a mean of 5.67 g and a standard deviation of 0.06 g. Some vending machines are designed so that you can adjust the weights of quarters that are accepted. If many counterfeit coins are found, you can narrow the range of acceptable weights with the effect that most counterfeit coins are rejected along with some legitimate quarters.

a. If you adjust vending machines to accept weights between 5.64g and 5.70 g, what percentage of legal quarters are rejected? Is that percentage too high?

b. If you adjust vending machines to accept all legal quarters except those with weights in the top 2.5% and the bottom 2.5%, what are the limits of the weights that are accepted?

Chapter 6-4 Exercise 10

Use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement.

a. For the population, find the proportion of odd numbers.

b. Table 6-3 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample proportion of odd numbers. Then combine values of the sample proportion that are the same, as in Table 6-4. (Hint: See Example 1 for Tables 6-4 and 6-4 that describe the sampling distribution of the sample mean.)

c. Find the mean of the sampling distribution of the sample proportion of odd numbers.

d. Based on the preceding results, is the sample proportion an unbiased estimator of the population proportion? Why or why not?

18: After constructing a new manufacturing mach9ine, five prototype integrated circuit chips are produced and it is found that two are defective (D) and three are acceptable (A). Assume that two of the chips are randomly selected with replacement from this population.

a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions of defects.

b. Find the man of the sampling distribution.

c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of defects? Does the mean of the sampling distribution of proportions always equal the population proportion?

Chapter 6-5 Exercise 14

According to the web site www.torchmate.com, “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey).

a. Use the multiplication rule to find the probability that your first two digits match those drawn and your last two digits do not match those drawn. That is, find P(MMXX), where M denotes a match and X denotes a digit that does not match the winning number.

b. Beginning with MMXX, make a complete list of the different possible arrangements of two matching digits and two digits that do not match, then find the probability for each entry in the list.

c. Based on the preceding results, what is the probability of getting exactly two matching digits when you select four digits for the Win 4 lottery game?

Chapter 5-4 Exercise 10

When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 580 peas, and Mendel theorized that 25% of them would be yellow peas.

a. If Mendel’s theory is correct, find the man and standard deviation for the numbers of yellow peas in such groups of 580 offspring peas.

b. The actual results consisted of 152 yellow peas. Is that result unusually high? What does this result suggest about Mendel’s theory?

16. The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter r is used at a rate of 6%.

a. Find the mean and standard deviation for the number of times the letter r will be found on a typical page of 2600 characters.

b. In an intercepted ciphered message sent to Iran, a page of 2600 characters is found to have the letter r occurring 178 times. Is this unusually low or high?

Chapter 6-2 Exercise 46

Draw a graph and find the probability of the given scores using table A-2 on page 249 or technology rounded to four decimal places:

About ________% of the area is between z = -2 and z = 2 (or within two standard deviations of the mean).

Chapter 6-3 Exercise 16

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Draw a graph in each case.)

16. Find the probability that a randomly selected adult has an IQ between 110 and 120 (referred to as a bright normal).

32. After 1964, quarters were manufactured so that he weights had a mean of 5.67 g and a standard deviation of 0.06 g. Some vending machines are designed so that you can adjust the weights of quarters that are accepted. If many counterfeit coins are found, you can narrow the range of acceptable weights with the effect that most counterfeit coins are rejected along with some legitimate quarters.

a. If you adjust vending machines to accept weights between 5.64g and 5.70 g, what percentage of legal quarters are rejected? Is that percentage too high?

b. If you adjust vending machines to accept all legal quarters except those with weights in the top 2.5% and the bottom 2.5%, what are the limits of the weights that are accepted?

Chapter 6-4 Exercise 10

Use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement.

a. For the population, find the proportion of odd numbers.

b. Table 6-3 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample proportion of odd numbers. Then combine values of the sample proportion that are the same, as in Table 6-4. (Hint: See Example 1 for Tables 6-4 and 6-4 that describe the sampling distribution of the sample mean.)

c. Find the mean of the sampling distribution of the sample proportion of odd numbers.

d. Based on the preceding results, is the sample proportion an unbiased estimator of the population proportion? Why or why not?

18: After constructing a new manufacturing mach9ine, five prototype integrated circuit chips are produced and it is found that two are defective (D) and three are acceptable (A). Assume that two of the chips are randomly selected with replacement from this population.

a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions of defects.

b. Find the man of the sampling distribution.

c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of defects? Does the mean of the sampling distribution of proportions always equal the population proportion?

Chapter 6-5 Exercise 14

According to the web site www.torchmate.com, “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey).

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