Question

*Problem #36) The problem numbers below correspond to problems in the textbook for section 6.2 (pg. 340). Use the scenario given in the textbook problem to answer the questions given here. In other words, replace parts (a), (b), etc. from the book with the questions below. Please show the probabilities necessary to support all of your answers, particularly when asked whether a particular result is unusual.

Problem #36) (a) Identify the properties of this binomial distribution: What does a "success" mean in this context? What is LaTeX: n n ? What is LaTeX: p p ? (b) What is the probability that exactly 20% of the people surveyed believe that moral values in the U.S. are poor? (Hint: What value of X does 20% correspond to?) (c) Would it be unusual for 80% of the people surveyed to believe that moral values in the U.S. are poor? (Careful... the question does not say "exactly 80%".)

distribill and standard deviation of the of Section 6.1; (c) compute the mean and standara using the methods of this section; and (d) draw a graph of the probability distribution and comment on its shape. experi (a) Fi si NW 29. n= 6, p = 0.3 30. n=8,p = 0.5 31. n= 9, p = 0.75 32. n=10,p = 0.2 (b) Fi a (c) F 33. n10, p = 0.5 34. n9, p = 0.8 (d) Applying the Concepts 39 Un ind NW 35. On-Time Flights According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 15 flights are randomly selected, and the number of on-time flights is recorded. Su as (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 10 flights are (a on time. (c) Find and interpret the probability that fewer than 10 flights are on time (d) Find and interpret the probability that at least 10 flights are on time (e) Find and interpret the probability that between 8 and 10 flights, inclusive, are on time. 36. Morality In a recent poll, the Gallup Organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings moral values in the United States on the overall state of (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor. (c) Find and interpret the probability that more than 16 of those surveyed feel the state of morals is poor. (d) Find and interpret the probability that 13 or 14 believe the state of morals is poor. (e) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values is poor in the United States? Why? 37. Toilet Flushing In the Healthy Handwashing Survey conducted hu found that 64%

Answer 1

Solution

Let X = number of people surveyed who believe that moral values in the U.S. are poor.

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and

p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ……....................................................……………..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST……..........…….(1a)

Mean (average) of X = E(X) = µ = np…………………..................................................………………...………..(2)

Variance of X = V(X) = σ² = np(1 – p)………………................................................……………………………..(3)

Standard Deviation of X = SD(X) = σ = √{np(1 – p)} ……..................................………………………………...(4)

Now, to work out the solution,

Part (a)

Properties of this binomial distribution:

Parameters: n = 25 and p = 0.45 Answer 1

Mean = 11.25 Answer 2

Standard deviation = 2.487 Answer 3

"success" in this context means: A person surveyed believes that moral values in the U.S. are poor. Answer 4

n = 25, p = 0.45 Answer 5

Part (b)

Probability exactly 20% of the people surveyed who believe that moral values in the U.S. are poor

= P(X = 5) [20% of 25 = 5]

= 0.0063 [vide (1a)] Answer 5

Part (c)

It would be unusual for 80% of the people surveyed to believe that moral values in the U.S. are poor. Answer 6

Rationale:

80% of 25 is 20.

P(X ≤ 19) = 0.9996 [vide (1a)], which is very high. Hence, the chance of 80% or more to believe that moral values in the U.S. are poor is very low.

DONE