Stats Unit 1 (Semester 2)
First you need to find the probability: 45/250 = .18. Then, you need to solve for , which allows you to find the z-score for the area: ; . So, p( < .18) = P(z < -.8) = .212. This result can be found in a table of standard normal probabilities or on your calculator by entering normalcdf(.8, 10). A. .02 B. .50 C. .40 D. You can't do a problem like this unless the population is at least 10 times the sample size. E. .09646
c. .40 (Description: Find the probabilities for 15 and 20: 15/80 = .1875, 20/80 = .25. This means you need to find P(.1875 < p-hat < .25). Start by finding the standard deviation. This allows you to find the z score for 15 while the score for 25 is 0. So, P(.1875 < p-hat < .25) = P(-1.302 < z < 0) = .40. This figure can be found using a table of standard normal probabilities or on your calculator by entering normalcdf(-1.302, 0).)
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) (Decription: With the continuity correction we don't want to include 20. Stop at the midpoint right below it, which is 19.5. The -E99 (-10 to the 99th power) is the lower bound of the area you're looking for; many other numbers (such as -1000) would work here, as long as they're several standard deviations below the mean.
Under what conditions can you use a normal distribution to approximate the binomial distribution?
np is equal to or greater than 10 and n(1 - p) is equal to or greater than 10
Consider a binomial event with B(75,.6). What expression represents the probability of getting exactly 50 successes?
(75 50)(.6)^50(.4)^25
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
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
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.
0.0047 (Description: Assuming p = .5, the mean would be np = 100 .5 = 50, and the standard deviation would be . The probability of getting 63 or higher in a normal distribution with mean 50 and standard deviation 5 is about .0047
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
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
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? A. .3759 B. .4601 C. .1587 D. You can't answer this question because the sample size is too small to allow us to use the Central Limit Theorem. E. .6241
A. .3759 Because the original population is normally distributed, we can assume that the sampling distribution has N(505,). You can find the solution on your calculator with normalcdf(510,1000,505,15.8 ) = .3759.
Which of the following is a list of common steps to inference? A. Identify the study, be sure the study and your sample are valid, calculate probabilities or confidence intervals, test for significance B. Draw a full distribution of convenience samples, draw a random sample, compare the two samples, test the difference between samples C. Draw a random sample, draw a second random sample, compare the two samples, test the difference between the samples D. Draw a random sample, calculate the mean, make a histogram, make a dot plot E. Draw a convenience sample, draw a random sample, compare the two samples, test the difference between the samples
A. Identify the study, be sure the study and your sample are valid, calculate probabilities or confidence intervals, test for significance
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. A. 25, 5 B. 25, 1.25 C. 5, 5 D. 5, 1.25 E. 25,
B. 25, 1.25
For the game of roulette, the mean winnings for one bet is approximately -.0526 with a standard deviation of about .9986. What's the probability that you come out ahead (win a positive amount) if you play 100 times? .4790 B. .2992 C. You can't win more than 50 games because the odds are against you. D. .7008 E. .5
B. .2992 (Description: The Central Limit Theorem applies. You can solve this on your calculator using normalcdf(0,1E99,-.0526, (.9986/square root of 100)).
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? A. 0.029 B. 0.669 C. 0.924 D. 0.147 E. 0.308
B. 0.669 This question asks for P(1st hit on 1st at bat) + P(1st hit on 2nd at bat) + P(1st hit on 3rd at bat). You can enter geometcdf(.308, 3) on your calculator to get the same result.
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? A. Answer will vary B. 5 C. 80% of the number of applicants D. 20% of the number of applicants E. 20
B. 5 Description: This is an average waiting-time problem. Since the probability of success is 20%, the average waiting time is 1/.2, or five applicants. Note that this is just the average number required if this activity were to be repeated many times.
Which of the following represents a geometric setting? A. The number of cards dealt from a deck before you get a 10 B. The number of people entering the intensive care unit at a particular hospital on any day C. The number red M & Ms in a handful of 25 M & Ms D. The number of random telephone numbers you dial until you get an answer E. The amount of time you wait in line at a bank before getting to the counter
D. The number of random telephone numbers you dial until you get an answer
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? A. binomcdf(100,.2,29) or binomCdf(100,.2,29) B. 1 - binompdf(100,.2,29) C. 1 - binomcdf(100,.2,30) or 1 - binomCdf(100,.2,30) D. binomcdf(100,.2,30) or binomCdf(100,.2,30) E. 1 - binomcdf(100,.2,29) or 1 - binomCdf(100,.2,29)
E. 1 - binomcdf(100,.2,29) or 1 - binomCdf(100,.2,29)
Which of the following are true? I. A sampling distribution of a statistic consists of all possible random samples of the same size from a given population. II. Regardless of the shape of the original population, for samples of size 2, . III. 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.
E. I, II, and III (Description: A sampling distribution for a statistic results when all possible values of that statistic are computed for random samples of the same size. The mean and standard deviation of a sampling distribution isn't affected by the sample size. Generally, samples of size 40 would be large enough for us to use the Central Limit Theorem.)
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? A. Ken Griffey Jr. will be able to get a hit by his 3rd at bat. B. Ken Griffey Jr. won't get a hit all 3 times at bat. C. Ken Griffey Jr. will try 3 battings. D. Ken Griffey Jr.'s 0.308 batting average changes significantly depending on the outcome of the previous at bat. E. Ken Griffey Jr.'s 0.308 batting average won't change (at least significantly) in each at bat.
E. Ken Griffey Jr.'s 0.308 batting average won't change (at least significantly) in each at bat. Description: The fact that it's late in the season means that one or two at bats won't change the probability of Griffey getting a hit (0.308) by much.
Which of the following indicates the value of (8 2)(.3)^2(.7)^6 ? binompdf(8,.7,2) or binomPdf(8,.7,2) B. binomcdf(8,.7,6) or binomCdf(8,.7,6) C. 1 - binompdf(8,.3,2) or 1 - binomPdf(8,.3,2) D. binomcdf(8,.3,2) or binomCdf(8,.3,2) done E. binompdf(8,.3,2) or binomPdf(n,p,x)
E. binompdf(8,.3,2) or binomPdf(n,p,x)
For which of these values of n and p, can you use the normal approximation to the binomial distribution? A. n = 100, p = .02 B. n = 10, p = .7 C. n = 15, p = 2/3 D. n = 20, p = .6 E. n = 60, p = .4
E. n = 60, p = .4
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? A. .00023 B. .0563 C. 0 D. We can't answer this question because the distribution is highly skewed. E. .2417
E. .2417 Even though the distribution is skewed, the sample size is large enough for us to use the Central Limit Theorem. You can solve this on your calculator with normalcdf(71,73,70,2) = .2417.
When rolling a six-sided die, what's the probability of having to roll six times before you get a 4? A. 0.665 B. 0.000107 C. 1 D. 0.167 E. 0.067
E. 0.067 In this scenario, the first five rolls won't give you a 4, but the sixth will. The calculation is (5/6)^5 X (1/6) = 0.067. You can use the TI-83 or TI-84 to find this answer by entering geometpdf(1/6, 6)
Which of the following is most important for accuracy in inferential statistics? A. A good calculator B. A normally distributed population C. A reputable source of funding D. Proper linear regression E. A properly drawn random sample
E. A properly drawn random sample
True or False: A B(225, 1/5) distribution can be approximated by an N(225, 1/5) distribution.
False (Description: In general, a B(n, p) distribution can be approximated by an distribution, which in this problem would be an N(45, 6) distribution)
True or False: The sampling distribution for samples of size 10, with p near .2, would be approximately normal.
False (Description: 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: 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 (description: 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.)
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? .317 B. .155 C. .212 D. .788 E. None of these
First you need to find the probability: 45/250 = .18. Then, you need to solve for , which allows you to find the z-score for the area: =.18 ; . So, p(p < .18) = P(z < -.8) = .212. This result can be found in a table of standard normal probabilities or on your calculator by entering normalcdf(.8, 10).
Which of the following are key questions in a test of statistical significance? I. If random chance is the only factor, what's the chance I'll get that result? II. Is my result so unlikely that something other than chance must be a factor? III. Am I asking a significant research question?
I and II
Which of the following are true? I. A binomial event has exactly two possible outcomes. II. If the population size is at least 20 times the sample size, the independence criteria for a binomial has been met. III. As long as the probability for each trial is clearly specified, different trials can have different probabilities of success.
I and II only
Which of the following are common to all cases of statistical inference? I. Sample statistics are used to estimate population parameters. II. Two means are compared. III. A sample proportion is used to estimate a population proportion.
I only
Which of the following are true of all sampling distributions? I. mean=mean, std. deviation = std. deviation/square root of N II.It is a probability distribution of a statistic. III. All samples must be the same size.
II and III only
Which of the following are not essential characteristics of a binomial event? I. Each outcome must be independent. II. The sample size must be at least 20. III. A trial can have only two possible outcomes.
II only
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: 0.35 Standard Deviation: 0.028
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: A geometric probability distribution is skewed.
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 (Description: The 10 samples represent all possible random samples of size two. Thus, by the definition of a sampling distribution, the 10 means for these samples would represent all possible values of the statistic of interest.)
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. Correct! 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.