AP Stats Extra Practice Chapter 6

According to New Jersey Transit, the 8:00 a.m. weekday train from Princeton to New York City has a 90% chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let 𝑌Y = the number of days on which the train arrives on time. What is the probability that the train arrived on time on fewer than 4 days?

0.0159

A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable 𝑌Y = money collected (in dollars) on a randomly selected ferry trip is shown here. Money collected0510152025Probability0.020.050.080.160.270.42 Find 𝑃(𝑌<20)P(Y<20). Interpret this result.

𝑃(𝑌<20)P(Y<20) = 0.31. There is a 0.31 probability that the amount of money collected on a randomly selected ferry trip is less than $20.

In an experiment on the behavior of young children, each subject is placed in an area with five toys. Past experiments have shown that the probability distribution of the number 𝑋X of toys played with by a randomly selected subject is as follows: Number of toys 𝑥𝑖xi012345Probability 𝑝𝑖pi0.030.160.300.230.17??? What is the probability that a randomly selected subject plays with at most 3 toys?

0.72

In which of the following situations would it be appropriate to use a Normal distribution to approximate probabilities for a binomial distribution with the given values of 𝑛n and 𝑝p?

n=100, p=0.2

Victoria parks her car at the same garage every time she goes to work. Because she stays at work for different lengths of time each day, the fee the parking garage charges on a randomly selected day is a random variable, 𝐺G. The table gives the probability distribution of 𝐺G. You can check that 𝜇𝐺=$14μG=$14 and 𝜎𝐺=$2.74σG=$2.74. In addition to the garage's fee, the city charges a $3 use tax each time Victoria parks her car. Let 𝑇T = the total amount of money she pays on a randomly selected day. Number of passengers 𝑔𝑖gi$10$13$15$20Probability 𝑝𝑖pi0.200.250.450.10 What is the value of 𝜇𝑇μT? Interpret this value.

𝜇𝑇μT = $17. If many, many days are randomly selected, the average amount of money that Victoria pays for parking will be about $17.

Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren't present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford's law. Call the first digit of a randomly chosen legitimate record 𝑋X for short. The probability distribution for 𝑋X is shown here (note that a first digit cannot be 0). First digit 𝑥𝑖xi123456789Probability 𝑝𝑖pi0.3010.1760.1250.0970.0790.0670.0580.0510.046 What is the expected value of 𝑋X? Interpret this value.

𝜇𝑋μX = 3.441. In many, many randomly chosen legitimate records, the average of the first digit would be about 3.441.

A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable 𝑌Y = money collected (in dollars) on a randomly selected ferry trip is shown here. Money collected0510152025Probability0.020.050.080.160.270.42 What is the mean of 𝑌Y? Interpret this value.

𝜇𝑌μY = $19.35. If many, many ferry trips are randomly selected, the average amount of money collected would be about $19.35.

A company has developed an "easystart" mower that cranks the engine with the push of a button. The company claims that the probability the mower will start on any push of the button is 0.9. Assume for now that this claim is true. On the next 30 uses of the mower, let 𝑇T = the number of times it starts on the first push of the button. What is the standard deviation of 𝑇T? Interpret this value.

𝜎𝑇=1.643σT=1.643. If many many rounds of 30 attempts were completed, the number of times the mower would start on the first push of the button would typically vary from the mean (27) by about 1.643.

Victoria parks her car at the same garage every time she goes to work. Because she stays at work for different lengths of time each day, the fee the parking garage charges on a randomly selected day is a random variable, 𝐺G. The table gives the probability distribution of 𝐺G. You can check that 𝜇𝐺μG = $14 and 𝜎𝐺=$2.74σG=$2.74. Number of passengers 𝑔𝑖gi$10$13$15$20Probability 𝑝𝑖pi0.200.250.450.10 What is the value of 𝜎𝑇σT? Interpret this value.

𝜎𝑇σT = $2.74. The total amount of money Victoria pays for parking will typically vary from the mean by about $2.74.

According to New Jersey Transit, the 8:00 a.m. weekday train from Princeton to New York City has a 90% chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let 𝑌Y = the number of days on which the train arrives on time. What is 𝑃(𝑌=4)P(Y=4)? Interpret this value.

(64)(0.90)4(0.10)2(64)(0.90)4(0.10)2. There is a 9.84% probability that exactly 4 of the 6 trains arrive on time.

According to financial records, 24% of U.S. adults have more debt on their credit cards than they have money in their savings accounts. Suppose that we take a random sample of 100 U.S. adults. Let 𝐷D = the number of adults in the sample with more debt than savings. What is the probability that 30 or more adults in the sample have more debt than savings? Use a Normal distribution to estimate this probability.

0.08

"As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, "Please try again!" while others said, "You're a winner!" The company advertised the promotion with the slogan "1 in 6 wins a prize." Alan decides to use a different strategy for the 1-in-6 wins game. He keeps buying one 20-ounce bottle of the soda at a time until he gets a winner." What is the probability that he buys exactly 5 bottles?

0.0804

A total of 11% of students at a large high school are left-handed. A statistics teacher selects a random sample of 100 students and records 𝐿L = the number of left-handed students in the sample. What is the probability that 15 or more students in the sample are left-handed? Use a Normal distribution to estimate this probability.

0.1006

According to financial records, 24% of U.S. adults have more debt on their credit cards than they have money in their savings accounts. Suppose that we take a random sample of 100 U.S. adults. Let 𝐷D = the number of adults in the sample with more debt than savings. What is the probability that 30 or more adults in the sample have more debt than savings?

0.1009

As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, "Please try again!" while others said, "You're a winner!" The company advertised the promotion with the slogan "1 in 6 wins a prize." Grayson's statistics class wonders if the company's claim holds true at a nearby convenience store. To find out, all 30 students in the class go to the store and each buys one 20-ounce bottle of the soda. What is the probability that two or fewer students would win a prize if the company's claim is true?

0.1028

Pedro drives the same route to work on Monday through Friday. His route includes one traffic light. According to the local traffic department, there is a 55% chance that the light will be red on a randomly selected work day. Suppose we choose 10 of Pedro's work days at random and let 𝑌Y = the number of times that the light is red. What is the probability that the light is red on exactly 7 days?

0.1665

"As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, "Please try again!" while others said, "You're a winner!" The company advertised the promotion with the slogan "1 in 6 wins a prize." Alan decides to use a different strategy for the 1-in-6 wins game. He keeps buying one 20-ounce bottle of the soda at a time until he gets a winner." What is the probability that he buys at most 6 bottles?

0.6651

Marti decides to keep placing a $1 bet on number 15 in consecutive spins of a roulette wheel until she wins. On any spin, there's a 1-in-38 chance that the ball will land in the 15 slot. How many spins do you expect it to take for Marti to win?

38

Billy likes to play cornhole in his free time. On any toss, he has about a 20% chance of getting a bag into the hole. As a challenge one day, Billy decides to keep tossing bags until he gets one in the hole. Does the scenario describe a geometric setting? If so, define an appropriate geometric random variable.

Assuming his shots are independent, 𝑋X = number of tosses needed to get a bag in the hole is a geometric random variable with 𝑝p = 0.20.

Continuous random variables A, B, and C all take values between 0 and 10. Their density curves, drawn on the same horizontal scales, are shown here. Rank the standard deviations of the three random variables from smallest to largest.

B, C, A

Put the names of all the students in your statistics class in a hat. Mix up the names, and draw 4 without looking. Let 𝑋X = the number whose last names have more than six letters. Does this scenario describe a binomial setting? Justify your answer.

No, this is not a binomial setting because the observations are not independent.

As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap. Some of the caps said, "Please try again!" while others said, "You're a winner!" The company advertised the promotion with the slogan "1 in 6 wins a prize." Grayson's statistics class wonders if the company's claim holds true at a nearby convenience store. To find out, all 30 students in the class go to the store and each buys one 20-ounce bottle of the soda. Two of the students in Grayson's class got caps that say, "You're a winner!" Does this result give convincing evidence that the company's 1-in-6 claim is false?

No. If the "1 in 6 wins" claim is true, there is a 10.28% probability that two or fewer students would win a prize. Because this outcome is not very unlikely, we do not have convincing evidence that the company's claim is false.

A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable 𝑌Y = money collected (in dollars) on a randomly selected ferry trip is shown here. Money collected0510152025Probability0.020.050.080.160.270.42 What is the median of 𝑌Y?

The median of 𝑌Y is $20.

Total gross profits 𝐺G on a randomly selected day at Tim's Toys follow a distribution that is approximately Normal with mean $560 and standard deviation $185. The cost of renting and maintaining the shop is $65 per day. Let 𝑃P = profit on a randomly selected day, so 𝑃=𝐺-65P=G-65. Describe the shape, center, and variability of the probability distribution of 𝑃P. MC Prompt: Which of the following gives the correct shape, center, and variability of the probability distribution of 𝑇T?

Shape: Approximately Normal, Center: $495, Variability: $185.

Victoria parks her car at the same garage every time she goes to work. Because she stays at work for different lengths of time each day, the fee the parking garage charges on a randomly selected day is a random variable, 𝐺G. The table gives the probability distribution of 𝐺G. You can check that 𝜇𝐺μG = $14 and 𝜎𝐺σG = $2.74. In addition to the garage's fee, the city charges a $3 use tax each time Victoria parks her car. Let 𝑇T = the total amount of money she pays on a randomly selected day. Garage fee 𝑔𝑖gi$10$13$15$20Probability 𝑝𝑖pi0.200.250.450.10 Which of the following correctly describes the shape of the graph of the probability distribution of 𝑇T? (not shown)

The distribution of total amount of money Victoria pays is roughly symmetric with a single peak at $18.

small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable 𝑌Y = money collected (in dollars) on a randomly selected ferry trip is shown. Money collected0510152025Probability0.020.050.080.160.270.42 Express the event "at least $20 is collected" in terms of 𝑌Y. What is the probability of this event?

The event is 𝑌≥20Y≥20. 𝑃(𝑌≥20)=0.69P(Y≥20)=0.69.

Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren't present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford's law. Call the first digit of a randomly chosen legitimate record 𝑋X for short. The histogram of the probability distribution for 𝑋X is shown here. What is the shape of this histogram?

The histogram shows a right-skewed distribution.

A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable 𝑌Y = money collected (in dollars) on a randomly selected ferry trip is shown here. 𝜇𝑌μY = $19.35. Money collected0510152025Probability0.020.050.080.160.270.42 Compare the mean and the median. Based upon the relationship between the mean and the median, what is the shape of the probability distribution?

The probability distribution is skewed to the left. This relationship makes sense because the mean of 𝑌Y is less than the median of 𝑌Y.

Ana is a dedicated Skee Ball player who always rolls for the 50-point slot. Ana's score 𝑋X on a randomly selected roll of the ball has the probability distribution shown here with mean 𝜇𝑋μX = 23.8 and standard deviation 𝜎𝑍σZ = 12.63. A player receives one ticket from the game for every 10 points scored. Define 𝑇T = number of tickets Ana gets on a randomly selected roll. Which of these statements about the shape of the probability distribution of 𝑇T is correct?

The probability distribution of 𝑇T has the same shape as the probability distribution of 𝑋X: skewed to the right.

Pedro drives the same route to work on Monday through Friday. His route includes one traffic light. According to the local traffic department, there is a 55% chance that the light will be red on a randomly selected work day. Suppose we choose 10 of Pedro's work days at random and let Y = the number of times that the light is red. Make a graph of the probability distribution of 𝑌Y. What is the shape of the probability distribution?

The shape of the probability distribution is roughly symmetric with a single peak at 𝑌=6Y=6.

A company has developed an "easystart" mower that cranks the engine with the push of a button. The company claims that the probability the mower will start on any push of the button is 0.9. Assume for now that this claim is true. On the next 30 uses of the mower, let 𝑇T = the number of times it starts on the first push of the button. Here is a histogram of the probability distribution of 𝑇T: Describe the shape of the probability distribution.

The shape of the probability distribution is skewed to the left with a single peak at 𝑇=27T=27 times.

A small ferry runs every half hour from one side of a large river to the other. The histogram of the probability distribution for the random variable 𝑌Y = money collected (in dollars) on a randomly selected ferry trip is shown here. What is the shape of this histogram?

The shape of the probability histogram is skewed to the left with a single peak at $25 collected.

Pedro drives the same route to work on Monday through Friday. His route includes one traffic light. According to the local traffic department, there is a 55% chance that the light will be red on a randomly selected work day. Suppose we choose 10 of Pedro's work days at random and let 𝑌Y = the number of times that the light is red. Is 𝑌Y a binomial random variable? Explain!

This is a binomial setting and 𝑌Y has a binomial distribution with 𝑛n = 10 and 𝑝p = 0.55.

Shuffle a standard deck of playing cards well. Then turn over one card at a time from the top of the deck until you get an ace. Does the scenario describe a geometric setting? If so, define an appropriate geometric random variable.

This is not a geometric setting because the trials are not independent.

To use a Normal distribution to approximate binomial probabilities, why do we require that both 𝑛𝑝np and 𝑛(1-𝑝)n(1-p) be at least 10?

When 𝑛𝑝≥10np≥10 and 𝑛(1-𝑝)≥10n(1-p)≥10, we know that the combination of 𝑛n and 𝑝p is such that the binomial distribution is close to Normal.

According to financial records, 24% of U.S. adults have more debt on their credit cards than they have money in their savings accounts. Suppose that we take a random sample of 100 U.S. adults. Let 𝐷D = the number of adults in the sample with more debt than savings. Can 𝐷D be modeled by a binomial distribution even though the sample was selected without replacement? Explain.

Yes, because 𝑛n = 100 is less than 10% of the size of the population (all U.S. adults).

According to New Jersey Transit, the 8:00 a.m. weekday train from Princeton to New York City has a 90% chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let 𝑌Y = the number of days on which the train arrives on time. Does this scenario describe a binomial setting? Justify your answer.

Yes, this is a binomial setting and 𝑌Y has a binomial distribution with 𝑛=6n=6 and 𝑝=0.90p=0.90.

In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. Should we use a binomial distribution to approximate this probability? Justify your answer.

Yes. Although she is sampling without replacement, the sample size (7) is less than 10% of the population size (100) and all other conditions are met.

A company has developed an "easystart" mower that cranks the engine with the push of a button. The company claims that the probability the mower will start on any push of the button is 0.9. Assume for now that this claim is true. On the next 30 uses of the mower, let 𝑇T = the number of times it starts on the first push of the button. Here is a histogram of the probability distribution of 𝑇T: Does 𝑇T have a binomial distribution? Justify your answer.

Yes. T has a binomial distribution with 𝑛n = 30 and 𝑝p = 0.90.

A total of 11% of students at a large high school are left-handed. A statistics teacher selects a random sample of 100 students and records 𝐿L = the number of left-handed students in the sample. Can 𝐿L be approximated by a Normal distribution? Explain.

Yes. The expected number of successes and the expected number of failures are both 10 or more so the large counts condition is met.

According to financial records, 24% of U.S. adults have more debt on their credit cards than they have money in their savings accounts. Suppose that we take a random sample of 100 U.S. adults. Let 𝐷D = the number of adults in the sample with more debt than savings. Can 𝐷D be approximated by a Normal distribution? Explain.

Yes. The expected number of successes and the expected number of failures are both 10 or more so the large counts condition is met.

In an experiment on the behavior of young children, each subject is placed in an area with five toys. Past experiments have shown that the probability distribution of the number 𝑋X of toys played with by a randomly selected subject is as appears in the given table. Number of toys 𝑥𝑖xi012345Probability 𝑝𝑖pi0.030.160.300.230.17??? Which of the following expresses the event "child plays with 5 toys" in terms of 𝑋X and gives the correct probability?

𝑃(𝑋=5)P(X=5) = 0.11

Marti decides to keep placing a $1 bet on number 15 in consecutive spins of a roulette wheel until she wins. On any spin, there's a 1-in-38 chance that the ball will land in the 15 slot. Would you be surprised if Marti won in 3 or fewer spins? Compute an appropriate probability to support your answer.

𝑃(𝑌≤3)=0.0769.P(Y≤3)=0.0769. Because the probability is not that small, I wouldn't be surprised if Marti won in 3 or fewer spins.

A company has developed an "easystart" mower that cranks the engine with the push of a button. The company claims that the probability the mower will start on any push of the button is 0.9. Assume for now that this claim is true. On the next 30 uses of the mower, let 𝑇T = the number of times it starts on the first push of the button. What is the mean of 𝑇T? Interpret this value.

𝜇𝑇=27μT=27. If many rounds of 30 attempts were completed, the average number of times the mower would start on the first push of the button would be about 27 times.