statistics ch 5 & 6

If calculations are​ time-consuming and if a sample size is no more than​ 5% of the size of the​ population, the​ _______ states to treat the selections as being independent​ (even if the selections are technically​ dependent).

5% Guideline for Cumbersome Calculations

Assume that when adults with smartphones are randomly​ selected, 56​% use them in meetings or classes. If 6 adult smartphone users are randomly​ selected, find the probability that at least 4 of them use their smartphones in meetings or classes. The probability is . 4618

Best way: P(x less than/equal to 4) = 1 - binomcdf(6,.56,3)

The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Complete parts​ (a) through​ (d) below. a. Find the probability of getting exactly 6 girls in 8 births. . 116. b. Find the probability of getting 6 or more girls in 8 births. . 13. c. Which probability is relevant for determining whether 6 is a significantly high number of girls in 8​ births: the result from part​ (a) or part​ (b)? d. Is 6 a significantly high number of girls in 8​ births? Why or why​ not? Use 0.05 as the threshold for a significant event.

C) The result from part​ b, since it is the probability of the given or more extreme result. D)​No, since the appropriate probability is greater than​ 0.05, it is not a significantly high number.

Which of the following is not a requirement of the binomial probability​ distribution?

The trials must be dependent.

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.

​Multiple-choice questions each have five possible answers left parenthesis a comma b comma c comma d comma e right parenthesis(a, b, c, d, e)​, one of which is correct. Assume that you guess the answers to three such questions. a. Use the multiplication rule to find ​P(WWC​), where C denotes a correct answer and W denotes a wrong answer. ​P(WWC​)equals=. 128 b. Beginning with WWC​, make a complete list of the different possible arrangements of one correct answer and two wrong answers​, then find the probability for each entry in the list. c. Based on the preceding​ results, what is the probability of getting exactly one correct answer when three guesses are​ made?

a) 1/5 or .2 probability CORRECT 4/5 or .8 probability INCORRECT P(WWC) = .8*.8*.2 = .128 b) ​P(WWC​)−see above ​P(WCW​)equals=. 128 ​P(CWW​)equals=. 128 c)P(WWC) = .128 + .128 + .128 = .384

Assume that hybridization experiments are conducted with peas having the property that for​ offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 38. Complete parts​ (a) through​ (c) below. a. Find the mean and the standard deviation for the numbers of peas with green pods in the groups of 38. The value of the mean is muμ=9.5 peas. The value of the standard deviation is sigmaσequals=2.7 peas. b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high. Values of 4.1 peas or fewer are significantly low. Values of 14.9 peas or greater are significantly high. c. Is a result of 33 peas with green pods a result that is significantly​ low? Why or why​ not? The result is significantly​ low, because 33 peas with green pods is less than 4.1 peas.

a) m = np = 38(.25) = 9.5 Square root of npq = sr (38)(.25)(1-.25) = 2.7 b)

Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a​ girl, but assume that the method has no​ effect, so the probability of a girl is 0.5. Assume that the groups consist of 40 couples. Complete parts​ (a) through​ (c) below. a. Find the mean and the standard deviation for the numbers of girls in groups of 40 births. The value of the mean is m μ=20. The value of the standard deviation is σ=3.2. b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high. Values of 13.6 girls or fewer are significantly low. Values of 26.4 girls or greater are significantly high. c. Is the result of 32 girls a result that is significantly​ high? What does it suggest about the effectiveness of the​ method?

a) μ = np μ = 40(.5) = 20 σ = Square root of npq (n = # of times a trial is repeated, p = probability, q = 1-p). σ = square root (40)(.5)(1-.5) = 3.2 b)Significantly low : values less than/= m-2σ 20 - 2(3.2) = 13.6 Significantly high: values greater than/= m + 2σ 20 + 2(3.2) = 26.4 c) The result is significantly​ high, because 32 girls is greater than 26.4 girls. A result of 32 girls would suggest that the method is effective.

Based on a​ poll, among adults who regret getting​ tattoos, 20​% say that they were too young when they got their tattoos. Assume that four adults who regret getting tattoos are randomly​ selected, and find the indicated probability. Complete parts​ (a) through​ (d) below. a. Find the probability that none of the selected adults say that they were too young to get tattoos. . 4096 b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos. . 4096 c. Find the probability that the number of selected adults saying they were too young is 0 or 1. . 8192 d. If we randomly select four ​adults, is 1 a significantly low number who say that they were too young to get​ tattoos?

a)EXACT binompdf(4, .2, 0) b)EXACT bonompdf(4, .2, 1) c) .4096 + .4096 d) No, because the probability that at most 1 of the selected adults say that they were too young is greater than 0.05.

Assume that when adults with smartphones are randomly​ selected, 46​% use them in meetings or classes. If 8 adult smartphone users are randomly​ selected, find the probability that exactly 5 of them use their smartphones in meetings or classes. The probability is . 1816.

binompdf(since exact) binompdf(trials{or n}, percent, x value {exact value}) binompdf(8,.46,5) = .1816

The accompanying table describes the random variable​ x, the numbers of adults in groups of five who reported sleepwalking. Complete parts​ (a) through​ (d) below. a. Find the probability of getting exactly 4 sleepwalkers among 5 adults. . 029 b. Find the probability of getting 4 or more sleepwalkers among 5 adults. . 033 c. Which probability is relevant for determining whether 4 is a significantly high number of sleepwalkers among 5​ adults: the result from part​ (a) or part​ (b)? d. Is 4 a significantly high number of 4 sleepwalkers among 5​ adults? Why or why​ not? Use 0.05 as the threshold for a significant event.

c) Since the probability of getting 5 sleepwalkers includes getting 4​ sleepwalkers, the result from part​ (b) is the relevant probability. d)​Yes, since the appropriate probability is less than​ 0.05, it is a significantly high number.

In the binomial probability​ formula, the variable x represents the​ _______.

number of successes.

Determine whether the given procedure results in a binomial distribution​ (or a distribution that can be treated as​ binomial). If the procedure is not​ binomial, identify at least one requirement that is not satisfied. 8 different senators from the current U.S. Congress are randomly selected without replacement and whether or not​ they've served over 2 terms is recorded. Does the probability experiment represent a binomial​ experiment?

​No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.

The accompanying table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children. Use the range rule of thumb to determine whether 1 girl in 10 births is a significantly low number of girls. Use the range rule of thumb to identify a range of values that are not significant. The maximum value in this range is 8.48.4 girls. The minimum value in this range is 1.61.6 girls. Based on the​ result, is 1 girl in 10 births a significantly low number of​ girls? Explain.

​Yes, 1 girl is a significantly low number of​ girls, because 1 girl is below the range of values that are not significant. Your answer is correct.