AP Statistics Semester 2 Quiz/Checkpoint Questions
A researcher collects infant mortality rate (IMR) data from a random sample of 200 villages in a large country. The mean IMR across these villages is 15.7 deaths per 1,000 infants born in a year. Based on previous research, the researcher assumes the population standard deviation is 6.5. What's the 90% confidence interval for the population IMR for villages in this country?
(14.94, 16.46)
A teacher administers a standardized math test to his class of 75 students. The mean score (out of 300 possible points) is 235. From previous studies, you know the population standard deviation is 28. Using the sample data given, calculate a 95% confidence interval for the population mean.
(228.7, 241.3)
Consider a binomial event with B(75, .6). Which of the following represents the probability of getting exactly 50 successes?
(75 50) (.6)^50 (.4)^25
In a blind ESP test, a person correctly identifies whether a tossed coin comes up heads or tails in 63 trials out of 100. Use the normal approximation (without the continuity correction) to calculate the probability of correctly identifying 63 or more.
.0047
Let x be a binomial random variable with n = 15 and p = .5. Using the exact binomial calculation and the normal approximation with the continuity correction, find P(X > 6).
.6964, .6972
A commercial crabber catches more than 1,000 crabs and measures the shells, and finds the mean length is 6.8 inches with a standard deviation of 3.2 inches. Assuming these measures are true for the population, if the crabber takes many random samples of size 50, what proportion of the sample means would we expect to be greater than 6 inches?
.9615 (P(X > 6) = P[z > (6 - 6.8) / (3.2 / sqrt 50)] = P(z > -1.77) = 0.9616)
What's your σ(sub x̅) value?
.968
You have a table of standard normal probabilities that gives you the area of the curve from the left tail to the z-score of interest. When using this type of table, what area of the curve would you use to find the corresponding z-score for confidence interval of 95%?
.975
Which of the following isn't necessary to compute the sample size appropriate for a given confidence level and margin of error?
Sample mean x-bar
Jeff is a snake charmer who can charm snakes with .78 probability of success. In a typical snake-charming season, Jeff attempts to charm 400 snakes. Using the exact binomial calculation, find the probability that he'll charm fewer than 300 snakes.
.067
If you flip three coins simultaneously, what's the probability you'll get three heads for the first time on the fifth toss?
.0733 (geometpdf(1/8, 5))
The probability that a given 80-year-old person will die in the next year is .27. What's the probability that exactly 10 of 40 80-year-olds will die in the next year?
.1385 (this is binompdf(40,.27,10))
Suppose you roll a six-sided die 10 times. What's the probability of getting three fives in those 10 rolls?
.155 (binompdf(10, 1/6, 3))
According to the manufacturer, the average proportion of red candies in a package is 20%. An 8 oz. package contains about 250 candies. What's the probability that a randomly selected 8 oz. bag contains less than 45 red candies?
.212
What's the probability of a sample of 10 students getting an average score of 510 or more on a standardized test if the test's scores are normally distributed with a mean of 505 and a standard deviation of 50?
.3759
About 25% of all dogs live more than 10 years. Out of a random sample of 80 dogs, what's the probability that between 15 and 20 dogs will live more than 10 years?
.40
In a survey, 1,000 mothers and fathers are asked about the importance of sports for boys and girls. Of the parents interviewed, 75% said genders are equal and should have equal opportunities to participate in sports. Assuming that .75 is correct for the population, what are the mean, standard deviation, and shape of the distribution of the sample proportion p hat for n = 100?
.75, .0433, approximately normal
Samples of size 49 are drawn from a distribution that's highly skewed to the right with a mean of 70 and a standard deviation of 14. What's the probability of getting a sample mean between 71 and 73?
0.02417
When rolling a 6-sided die, what's the probability of having to roll 6 times before you get a 4?
0.067 (geometpdf(1/6, 6))
If you flip two coins simultaneously, what's the probability you'll have to flip them four times before the first occurrence of two heads?
0.11 (geometpdf(1/4, 4))
Suppose you flip a coin n times, and the probability of getting heads 15 times is .0148. What's n?
20 (If you flip a coin 20 times, there's a .0148 probability that you'll get 15 heads. You can solve this in one of two ways: 1. Use the binomial formula, plug in the values you know for X, p, andP open parentheses X equals 15 close parentheses, and solve for n. 2. Using binompdf on your calculator, try all answer choices and choose the one that gives you a probability of .0148.)
Which of these represents the probability of getting doubles (getting the same number on two dice) on or before the seventh roll of two six-sided dice?
P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) (this equals geometcdf(1/6, 7))
In a blind ESP test, a person correctly identifies whether a tossed coin comes up heads or tails in 63 trials out of 200. Using the normal approximation (without the continuity correction), which of the following would you use to calculate the probability of correctly identifying 63 or more?
P(z > -5.233)
Frank is a champion fish charmer who can charm fish with 0.62 probability of success. If he attempts to charm 100 fish, what's the probability he'll charm between 60 and 80 inclusive? Solve using two methods: normal approximation without the continuity correction, and normal approximation with the continuity correction.
0.660, 0.697
Suppose you attend a baseball game late in the season. The Seattle Mariners are playing, and Ken Griffey Jr.'s batting average is 0.308. What's the probability he'll get his first hit of the game on or before his 3rd at bat?
0.669 (geometcdf(.308, 3))
If you roll two dice, what's the probability of rolling a seven (the numbers on the dice add up to 7) on or before the eighth roll?
0.767 (geometcdf(1/6, 8))
What are the mean and the standard deviation of a sampling distribution consisting of samples of size 16? These samples were drawn from a population whose mean is 25 and whose standard deviation is 5.
25, 1.25
Jeff typically makes three out of nine attempted free throws. What's the average waiting-time for Jeff to make his first basket, and what's the probability he'll make a basket on or before the very last attempt within his average waiting-time?
3, 0.70 (1/.3 = 3, and geometcdf(1/3, 3))
You take a 100-question multiple-choice test. Each question has five choices, and you guess at each question. Which of the following calculator commands would give you the probability of getting at least 30 questions correct?
1 - binomcdf(100,.2,29) (Binomcdf(n, p, x) will sum the individual probabilities from X = 0 through X = x. Since you want the sum of all possibilities starting at 30, you must subtract the sum of all possibilities less than 30 from 1.)
Which of the following are key questions in a test of statistical significance? 1. If random chance is the only factor, what's the chance I'll get that result? 2. Is my result so unlikely that something other than chance must be a factor? 3. Am I asking a significant research question?
1 and 2
Which of the following are true? 1. A binomial event has exactly two possible outcomes. 2. If the population size is at least 20 times the sample size, the independence criteria for a binomial has been met. 3. As long as the probability for each trial is clearly specified, different trials can have different probabilities of success.
1 and 2 only
Which of the following are common to all cases of statistical inference? 1. Sample statistics are used to estimate population parameters. 2. Two means are compared. 3. A sample proportion is used to estimate a population proportion.
1 only
Which of the following are true of all sampling distributions? 1. μ = μ and σ = σ/sqrt(n) 2. It is a probability distribution of a statistic. 3. All samples must be the same size.
2 and 3 only
Which of the following are not essential characteristics of a binomial event? 1. Each outcome must be independent. 2. The sample size must be at least 20. 3. A trial can have only two possible outcomes.
2 only
What's your value for z*?
2.17
You compute a 99% confidence interval with sample of size A and a margin of error of + or - 5 units. You now wish to compute a 90% confidence interval with the same margin of error with a sample of size B. For this situation, which of the following is true?
A is greater than B. (To achieve the same margin of error (that is, 5 units) at a lower confidence level, it's not necessary for your sample to be as large as it was for a higher confidence level.)
Which of the following situations satisfies all the conditions of a binomial setting? (These conditions are: we know the number of repetitions, the outcome of each trial can be considered either a success or a failure, we know the probability of success or failure of any trial, and the probability doesn't change from trial to trial.)
A jar contains 500 balls-300 red and 200 white. Ten balls are randomly selected from the jar, and the number X of red balls is recorded.
Which of the following is most important for accuracy in inferential statistics?
A properly drawn random sample
If a success is defined as getting a three on a six-sided die, what's P(x < 3) if you roll the die 10 times?
binomcdf(10,1/6,2)
A twenty-sided die, with the faces numbered 1 to 20, is rolled 100 times. What's the exact probability of getting more than 15 elevens but at most 30 elevens?
binomcdf(100, 1/20, 30) - binomcdf(100, 1/20, 15)
What's the probability of getting exactly 1 five when we roll a fair die 10 times?
binompdf(10, 1/6, 1) = .323
Which of the following indicates the value of (8 2) (.3)^2 (.7)^6?
binompdf(8,.3,2)
True or False: For a population whose mean is 100 and whose standard deviation is 15, 1000 random samples of size 20 are enough to generate a sampling distribution
false (1000 random samples may give you a very good simulation of the sampling distribution, but the sampling distribution is composed of all possible random samples of a given size. To emulate a sampling distribution we use either simulations of sampling distributions or the laws of probability.)
For the following 4 questions, refer to this data: To estimate the mean height of female high school juniors, you take a random sample of 30 female students and get these results (in inches): 72, 51, 67, 68, 61, 69, 58, 56, 60, 56 66, 61, 60, 59, 59, 54, 58, 53, 68, 63 57, 62, 63, 64, 56, 62, 58, 67, 57, 70 If the σ is 5.3, based on past research, and you want to construct a 97% confidence interval using (estimate) ± (margin of error) = (estimate) ± z*(σ/√n), what's your point estimate of µ?
61.17
Which statement best describes the meaning of the term average waiting-time?
The average number of trials required to get the first success
Using a random sample of 2,000 students, you compute a 95% confidence interval to estimate the mean calories consumed by eighth graders. You decide to compute another 95% confidence interval using a different sample, this time with only 1,000 students. What change would you expect from the first confidence interval to the second?
The confidence interval will be wider.
True or False: A 95% confidence interval is narrower than a 90% confidence interval for the same data set.
false (A 95% confidence interval is wider than a 90% confidence interval because we need to include more possible values in our estimate to increase our confidence that we've captured the population mean.)
True or False: The sampling distribution for samples of size 10, with p near .2, would be approximately normal.
false (Since np = (10)(.2) = 2 < 10, the assumptions needed to use the normal approximation aren't satisfied-np and n(1 - p) must both be greater than 10. (You might have a textbook that says that np and n(1 - p) need to be greater than or equal to 5, but this Tutorial follows the standard that they should be greater than or equal to 10.))
True or False: A B(225, 1/5) distribution can be approximated by an N(225, 1/5) distribution.
false (it would be an N(45, 6) distribution)
Suppose you're playing a game where you roll two dice. If you get doubles on or before the fourth roll you win the game. What's the probability of winning?
geometcd(1/6, 4)
You have an SRS of 300 students selected from over 100,000 college students. Of your sample, 35% said they had fallen asleep in their English class at least once during the previous semester. The mean and standard deviation of this statistic are:
mean = .35 and standard deviation = .028
After choosing a simple random sample from a population having unknown mean µ and an estimated standard deviation σ, you use the following formula to construct a confidence interval for µ: ± z*(σ/√n). In this formula, which of the following is represented by ± z*?
the z-score equivalent for the confidence interval of µ
True or False. If we take a properly drawn sample and calculate the mean and standard deviation, we can estimate the mean of the population and we can come up with a probability that the true mean falls within a certain interval.
true
True or False: Given that there are only 10 different possible samples of size two that can be selected from a population of five values, the sampling distribution of the mean would be composed of the means of these 10 samples.
true
A random sample of 85 adults found that average calorie consumption was 2,100 per day. Previous research has found a standard deviation of 450 calories, and you use this value for . Construct a 99% confidence interval for the population mean.
(1,974.3, 2,225.7)
What's this interval?
(59.07, 63.27)
As part of a promotional campaign, a gas station hands out game pieces to customers at a rate of one per visit. Each piece has a letter or an exclamation point on it, and if you collect all seven game pieces they spell you win! If you've collected six of the seven game pieces, and all you need is the letter w to get all seven, what's the probability you'll get your piece on the fifth trip to the gas station? Assume there are equal numbers of each piece.
(6/7)^4(1/7)
A researcher is interested in estimating the mean blood alcohol content (BAC) of people arrested for driving under the influence. The sample consists of 250 individuals with a mean BAC of .145. Based on past data, the researcher assumes a population standard deviation of .065. What's the 95% confidence interval for this scenario?
(.137, .153)
In a deck of 52 playing cards, there are 12 face cards (4 Jacks, 4 Queens, 4 Kings). If you draw cards one at a time, replacing and shuffling the cards between draws, what's the probability of getting your first face card on the third draw? Choose the best answer.
(1 - 3/13)^2(3/13)
If you roll a six-sided die 10 times, what's P(x > 3)?
1-binomcdf(10,1/6,3)
What's the margin of error?
2.1 inches
The following frequency plots represent observed frequency data gathered from probability experiments. Which of these indicates an experiment in a geometric setting?
(Note that this doesn't look exactly like a plot of a geometric distribution. That's because it's a plot of observed values, not theoretical ones. If you ran the experiment for a large enough number of trials, you'd see a regular geometric distribution, with the highest frequency at n = 1. Since the probability for each successive n is a fraction of the previous n, the probabilities will continue to get smaller as n gets larger.)
The probability that a given 80-year-old person will die in the next year is .27. What's the probability that between 10 and 15 (inclusive) of 40 80-year-olds will die in the next year?
..6191 (binomcdf(40,.27,15)
A researcher is interested in estimating the mean blood alcohol content (BAC) of people arrested for driving under the influence. The sample consists of 250 individuals with a mean BAC of .145. Based on past data, the researcher assumes a population standard deviation of .065. What's the margin of error for a 90% confidence interval in this scenario?
.0068
For the game of roulette, the mean winnings for one bet is approximately minus 0.0526 with a standard deviation of about 0.9986. What's the probability that you come out ahead (win a positive amount) if you play 100 times?
0.2992
A bag of candy has equal numbers of candies in eight colors: blue, red, brown, green, yellow, orange, pink, and black. If you eat them one by one, what's the probability of getting your first red candy on or before the fifth pick?
0.487 (geometcdf(1/8, 5)
Eric is a statistician-Viking. He walks into a tavern and makes three statements about the continuity correction. Which are true? 1. You can find single probabilities (such as x = 4) using the normal approximation to the binomial. Do this by finding the probability for the range from .5 below to .5 above the number. (For x = 4 you'd find 3.5 < x < 4.5) 2. Using the normal approximation, x < 4 is the same as x ≤ 4. Using the exact binomial, x < 4 is not the same as x ≤ 4. 3. The continuity correction removes some of the error introduced when modeling a discrete distribution with a continuous one. The correction increases accuracy by acting as though a number occupies the interval from 0.5 below it to 0.5 above it.
1, 2, and 3
Which of the following are true? 1. A sampling distribution of a statistic consists of all possible random samples of the same size from a given population. 2. Regardless of the shape of the original population, for samples of size 2, μ = μ and σ = σ/sqrt(n). 3. Unless there was extreme skewness or outliers, we can assume that a sampling distribution of a sample mean was approximately normal for samples of size 40.
1, 2, and 3
A researcher collects infant mortality rate (IMR) data from a random sample of 200 villages in a large country. The mean IMR across these villages is 15.7 deaths per 1,000 infants born in a year. Based on previous research, the researcher assumes the population standard deviation is 6.5. Then the researcher decides to take a larger sample so he can estimate the mean IMR with a margin of error of .5 infants per 1,000 born in a year at a 99% confidence interval. How many villages would the researcher have to sample?
1,122 (1,121 was wrong; round up)
What's z* for a 78% confidence interval?
1.22
What's the critical z-value for an 85% confidence interval?
1.44
In a binomial distribution with sample size n = 65, and probability of success p = .8, what would the approximate mean of the distribution be?
52
A teacher administers a standardized math test to his class of 75 students. The mean score (out of 300 possible points) is 235. From previous studies, you know the population standard deviation is 28. The principal has decided that she wants to estimate the average score to within 4 points (margin of error = 4) with 99% confidence. If she can only administer the test to one random sample of students, how large should this sample be to achieve the desired margin of error and confidence level?
326 (Since you can't sample part of a student, round up your sample size to the next whole number. (325 was wrong))
The standard deviation of SAT scores is 100 points. A researcher decides to take a sample of 500 students' scores to estimate the mean score of students in your state. What is the standard deviation of the sample mean?
4.47
Suppose population data suggests that 20% of applicants to a statistical surveying job will have prior surveying experience. How many candidates would have to be interviewed, on average, to find someone with prior surveying experience?
5 (1/.2)
In a binomial distribution with n = 140 and p = .62, what is the expected standard deviation of the distribution, to the nearest hundredth?
5.74
What's the probability of getting 1 or 3 fives on 10 rolls of a fair die? Hint: Remember the rule for finding P(A or B).
B(10, 1/6, 1) + B(10, 1/6, 3) (P(X = 1) is binompdf(10, 1/6, 1) and P(x = 3) is binompdf(10, 1/6, 3). To combine the two probabilities using or, simply add them: P(X = 1) + P(X = 3) = .323 + .155 = .478.)
Fred is a weightlifter who can lift 800 pounds on 45% of his attempts. Which of these expressions represents the probability Fred will make 30 lifts out of 60?
B(60, .45, 30)
For which of these scenarios could you use the normal approximation to the binomial?
Flip a coin 100 times and count the number of heads.
Which of the following is a list of common steps to inference?
Identify the study, be sure the study and your sample are valid, calculate probabilities or confidence intervals, test for significance
The 99.7% confidence interval for the mean length of frog jumps is (12.64 cm, 14.44 cm). Which of the following statements is a correct interpretation of 99.7% confidence? - Of the total number of frogs in your area of the country, 99.7% can jump between 12.64 cm and 14.44 cm. - There's a 99.7% chance that the mean length of frog jumps falls between 12.64cm and 14.44 cm. - If we were to repeat this sampling many times, 99.7% of the confidence intervals we could construct would contain the true population mean. - 99.7% of the confidence intervals we could construct after repeated sampling would go from 12.64 cm to 14.44 cm. - There's a 99.7% chance that any particular frog I catch can jump between 12.64 cm and 14.44 cm.
If we were to repeat this sampling many times, 99.7% of the confidence intervals we could construct would contain the true population mean.
A researcher computes a 90% confidence interval for the mean weight (in lbs) of widgets produced in a factory. The interval is (7.2, 8.9). Which of these is a correct interpretation of this interval?
If you drew many samples of size n and constructed a confidence interval from each sample, 90% of the intervals would contain the true population value.
You're at a Seattle Mariners baseball game late in the season, when Ken Griffey Jr.'s batting average is 0.308. You want to calculate the probability, using a binomial setting, that he'll get his first hit of the game on or before his third at bat. What assumption(s) do you have to make to get your answer?
Ken Griffey Jr.'s 0.308 batting average won't change (at least significantly) in each at bat.
Out of a population of 10,000 voters, 60% vote for Carl Fredrick Gauss. You take many, many samples of size 30 from this population and, for each sample, count the number of people who vote for Gauss. What's the approximate distribution of counts of votes for Gauss?
N(18, 2.68)
The B(225, 1/5) distribution can be approximated by what normal distribution?
N(45, 6)
Joe is a fish thrower who throws fish into a chute at a processing plant. Joe misses the chute 20% of the time. Using the normal approximation with the continuity correction, which of these correctly represents the probability Joe will miss fewer than 80 times if he throws 500 fish?
P(X < 79.5) for N(100, 8.94)
A friend of yours can shoot free throws with 70% accuracy (she makes 70% of her shots). If she attempts 25 free throws, what's the probability she'll make fewer than 14?
P(x < 14)
A friend of yours can shoot free throws with 75% accuracy (she makes 75% of her shots). If she attempts 25 free throws, what's the probability she'll make at least 20?
P(x = 20) + P(x = 21) + P(x = 22) + P(x = 23) + P(x = 24) + P(x = 25)
Suppose voters from a simple random sample of 500 (N > 1,000,000) are interviewed and asked which presidential candidate they're going to vote for. Of these, 35% say they'll vote for the Statistics Party. You want to know the probability, assuming this proportion is correct for the population, that more than 40% of a random sample of 500 people will vote for the Statistics Party. Which of these shows the three ways you could find your answer?
Proportion, using normal approximation: normalcdf(.40, E99, .25, .0213) Exact binomial count: 1 - binomcdf(500, .35, 200) Binomial count, using normal approximation: normalcdf(200, E99, 17, 10.66)(200, E99, 175, 10.66)
Which of the following represents a geometric setting?
The number of random telephone numbers you dial until you get an answer (This meets the criteria for a geometric setting: each trial has just two outcomes (usually success and failure), and the probability of success is the same for each trial (usually referred to as p). In the geometric situation, however, the random variable X of interest is the number of trials required to obtain the first success. In this case, someone either answers or doesn't, and the probability p of success should be the same each time.)
In the geometric setting the trials are independent, each trial has just two possible outcomes (success and failure), the probability of success is the same for each trial (referred to as p), and the random variable X is the number of trials required to get the first success. Which of the following scenarios meets the requirements of a geometric setting?
There are 10 different prizes in boxes of Googily-Snaps. Prize four is the most valuable among collectors. What's the probability that I'll get prize four without having to buy more than 4 boxes? (Each trial is independent, each trial has the same probability of success or failure, and you're interested in the probability of success within a certain number of trials.)
The sample mean, x̅, is called a __________ of the population mean µ.
point estimate
For which of these values of n and p, can you use the normal approximation to the binomial distribution?
n = 60, and p = .4
The probability of finding a mistake on an Income Tax Return is about .23. An employee of the IRS plans to inspect 100 random returns. Using the normal approximation to the binomial and the continuity correction, you want to calculate the probability that the workers find less than 20 mistakes. Which matches most closely what you'll enter into your calculator?
normalcdf(-E99, 19.5, 23, 4.21)
John is an expert horseshoe thrower who only misses 15% of the time. Choose the expression that correctly represents the probability John will miss fewer than 50 times if he throws 400 horseshoes.
normalcdf(-E99, 50, 60, 7.14)
Joe is a fish thrower who throws fish into a chute at a processing plant. Joe misses the chute 20% of the time. Using the normal approximation with the continuity correction, how would you calculate the probability Joe will miss fewer than 80 times if he throws 500 fish?
normalcdf(-e99, 79.5, 100, 8.94)
Under what conditions can you use a normal distribution to approximate the binomial distribution?
np ≥ 10 and n(1 - p) ≥ 10
True or False: The following situation could be considered a binomial experiment: In 1999, 25,000 students took the AP Statistics Exam. The probability that a randomly selected student from this group passed the exam was about .6. A statistician wants to know the likelihood that more than 650 out of 1000 students randomly selected from this group passed the exam.
true (Even though this situation doesn't strictly fit the definition of a binomial event (there are no independent trials), it's still a binomial event, since the population is more than 20 times greater than the sample size.)
True or False: A geometric probability distribution is skewed.
true (Geometric distributions are always skewed, though some are less skewed than others. There's a limit on one end and a long tail of diminishing probabilities on the other end.)
True or False: Increasing the sample size will decrease the margin of error in your confidence interval.
true (Larger sample sizes reduce variability, so your estimate will be more accurate and your margin of error will be smaller.)
