sta midterm ch5
Ten peas are generated from parents having the green/yellow pair of genes, so there is a 0.75 probability that an individual pea will have a green pod. Find the probability that among the 10 offspring peas, at least 9 have green pods. Is it unusual to get at least 9 peas with green pods when 10 offspring peas are generated? Why or why not?
0.244 explanation: 10P9 = (10!/(10-9)!9!)(.75^9)(.25^10-9) =.1877117157 10P10 = (10!/(10-10)!10!)(.75^10)(.25^10-10) =.0563135147 = .1877117157 + .0563135147 =.244 No, because the probability of this occurring is not small.
A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 23 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If a particular shipment of thousands of aspirin tablets actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted?
0.766 P(x=1) + P(x=0) = (23!/(23-1)!1!)(0.04^1)(.96^22) = .3747614876 = (23!/(23-0)!0!)(.04^0)(.96^23) =.3910554653 =.3910554653 + .3747614876 = .766
binomial probability distribution requirements
1. The procedure has a fixed number of trials 2. The trials must be independent 3. Each trial must have all outcomes classified into two categories 4. The probability of a success remains the same in all trials
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Three cards are selected from a standard 52-card deck without replacement. The number of diamonds selected is recorded.
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Determine whether or not the procedure described below results in a binomial distribution. If it is not binomial, identify at least one requirement that is not satisfied. Four hundred different voters in a region with two major political parties, A and B, are randomly selected from the population of 7000 registered voters. Each is asked if he or she is a member of political party A, recording Yes or No.
No, the trials are not independent and the sample is more than 5% of the population.
Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. n=7, x=2, p=0.25 P(2)=
P(2)= .311 (7!/(7-2)!2!)(0.25^2)(0.75^7-2) =.311
Binomial Probability Formula
P(x)= (nCx or n!/(n-x)!x!) (p^x) (q^n-x)
If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. This represents the _______.
Rare Event Rule
Five males with a particular genetic disorder have one child each. The random variable x is the number of children among the five who inherit the genetic disorder. Determine whether the table describes a probability distribution. If it does, find the mean and standard deviation. x P(x) 0 0.1804 1 0.3685 2 0.3010 3 0.1229 4 0.0251 5 0.0021
The table describes a probability distribution, it adds up to 1 and all items are between 0 and 1. mean = 1.5 explanation: sum of [x(P(x))]= 1.5 standard deviation = 0.9 explanation: the square root of {the sum of [x^2(P(x))] - mean^2} = 0.9
Determine whether the given procedure results in a binomial distribution. If it is not binomial, identify the requirements that are not satisfied. Randomly selecting 50 adults and asking if they like rap music, recording Yes or No
Yes because all four requirements are satisfied.
A Gallup poll of 1236 adults showed that 12% of the respondents believe that it is bad luck to walk under a ladder. Consider the probability that among 30 randomly selected people from the 1236 who were polled, there are at least 2 who have that belief. Given that the subjects surveyed were selected without replacement, the events are not independent. Can the probability be found by using the binomial probability formula? Why or why not?
Yes. Although the selections are not independent, they can be treated as being independent by applying the 5% guideline.
A TV show, Lindsay and Tobias, recently had a share of 25, meaning that among the TV sets in use, 25% were tuned to that show. Assume that an advertiser wants to verify that 25% share value by conducting its own survey, and a pilot survey begins with 14 households having TV sets in use at the time of a Lindsay and Tobias broadcast. a. Find the probability that none of the households are tuned to Lindsay and Tobias. b. Find the probability that at least one household is tuned to Lindsay and Tobias. c. Find the probability that at most one household is tuned to Lindsay and Tobias. d. If at most one household is tuned to Lindsay and Tobias, does it appear that the 25% share value is wrong? Why or why not?
a. 0.018 n=14 x=0 (because none) p=.25 = (14!/(14-0)!0!)(.25^0)(.75^14) = .018 b. 0.982 = P(X>=1)= 1 - P(x=0) = 1 - .018 = .982 c. 0.101 P(X=0) + P(X=1) = (14!/(14-1)!1!)(.25^1)(.75^14-1) = .0831504241 = .0831504241 + 0.018 = .101 d. No, because with a 25% rate, the probability of at most one household is greater than 0.05.
The accompanying table describes results from eight offspring peas. The random variable x represents the number of offspring peas with green pods. a. Find the probability of getting exactly 7 peas with green pods. b. Find the probability of getting 7 or more peas with green pods. c. Which probability is relevant for determining whether 7 is an unusually high number of peas with green pods, the result from part (a) or part (b)? d. Is 7 an unusually high number of peas with green pods? Why or why not? Use 0.05 as the threshold for an unusual event.
a. 0.223 (find the corresponding probability in the table given for 7) b. 0.334 (find the corresponding probabilities in the table given for 7 and all of the numbers above, add them) c. part b (7 or more) d. no, the probability is greater than 0.05
In a state's Pick 3 lottery game, you pay $1.48 to select a sequence of three digits (from 0 to 9), such as 311. If you select the same sequence of three digits that are drawn, you win and collect $418.28. a. How many different selections are possible? b. What is the probability of winning? c. If you win, what is your net profit? d. Find the expected value? e. If you bet $1.48 in a certain state's Pick 4 game, the expected value is −$1.06. Which bet is better, a $1.48 bet in the Pick 3 game or a $1.48 bet in the Pick 4 game? Explain.
a. 1000 10^3 because 3 digits b. 0.001 or 1/1000 c. $416.80 418.28 - 1.48 = $416.80 d. -1.06 explanation: e = sum of (x)(P(x)) -1.48(.999) + 416.8(.001) = -1.06 e. Neither bet is better because both games have the same expected value.
Determine whether the following value is a continuous random variable, discrete random variable, or not a random variable. a. The square footage of a pool b. The hair color of adults in the United States c. The number of free dash throw attempts before the first shot is missed d. The time it takes for a light bulb to burn out e. The number of people with blood type B in a random sample of 14 people f. The time required to upload a file to the Internet
a. continuous b. not a random variable c. discrete d. continuous e. discrete f. continuous
Determine whether the value is a discrete random variable, continuous random variable, or not a random variable. a. The number of statistics students now reading a book b. The exact time it takes to evaluate 27+72 c. The response to the survey question "Did you smoke in the last week?" d. The number of fish caught during a fishing tournament e. The time required to download a file from the Internet f. The number of hits to a website in a day
a. discrete b. continuous c. not a random variable d. discrete e. continuous f. discrete
Several psychology students are unprepared for a surprise true/false test with 17 questions, and all of their answers are guesses. a. Find the mean and standard deviation for the number of correct answers for such students. b. Would it be unusual for a student to pass by guessing (which requires getting at least 14 correct answers)? Why or why not?
a. μ= 8.5 σ=2.1 b. Yes, because 14 is greater than the maximum usual value. explanation: max value μ+2σ 8.5 + 4.2 = 12.7 < 14
In a probability histogram, there is a correspondence between _______.
area and probability
A _______ random variable has infinitely many values associated with measurements.
continuous
A _______ random variable has either a finite or a countable number of values.
discrete
expected value
e = sum of (x)(P(x))
The _______ of a discrete random variable represents the mean value of the outcomes.
expected value
In a clinical trial of a cholesterol drug, 241 subjects were given a placebo, and 12% of them developed headaches. For such randomly selected groups of 241 subjects given a placebo, identify the values of n, p, and q that would be used for finding the mean and standard deviation for the number of subjects who develop headaches.
n=241 p=.12 q=.88
Identify the expression for calculating the mean of a binomial distribution.
np
In the binomial probability formula, the variable x represents the _______.
number of successes
A _______ variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.
random
A main goal in statistics is to interpret and understand the meaning of statistical values. The _______ can be very helpful in understanding the meaning of the mean and standard deviation.
range rule of thumb
For the binomial distribution, which formula finds the standard deviation?
square root of npq
Which of the following is not a requirement of the binomial probability distribution?
the trials must be dependent
Which of the following is NOT one of the three methods for finding binomial probabilities that is found in the chapter on discrete probability distributions?
use a simulation
Assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p. Use the given values of n and p to find the mean μ and standard deviation σ. Also, use the range rule of thumb to find the minimum usual value μ−2σ and the maximum usual value μ+2σ. n=1472 p=1/2
μ= 736 explanation: μ=np σ= 19.2 explanation: σ=square root of npq min val= 697.6 explanation: μ−2σ Max val= 774.4 explanation: μ+2σ
