Questions: 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. If a survey of a random sample of 20 adults in this country is conducted in which they are asked to disclose their feelings on the overall state of moral values, complete parts (a) through (g) below. E. The probability of success is the same for each trial of the experiment. F. The trials are independent. G. The probability of success is different for each trial of the experiment. H. There are two mutually exclusive outcomes, success or failure. (b) Using the binomial distribution, determine the values of n and p. n=20 (Type an integer or a decimal. Do not round.) p=0.45 (Type an integer or a decimal. Do not round.) (c) Using the binomial distribution, find and interpret the probability that exactly 12 of those surveyed feel the state of morals is poor. The probability that exactly 12 of those surveyed feel the state of morals is poor is 0.0727. (Round to four decimal places as needed.) Interpret the results. In 100 trials of this experiment, we expect about to result in exactly 12 adults who feel the state of morals is

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. If a survey of a random sample of 20 adults in this country is conducted in which they are asked to disclose their feelings on the overall state of moral values, complete parts (a) through (g) below. E. The probability of success is the same for each trial of the experiment. F. The trials are independent. G. The probability of success is different for each trial of the experiment. H. There are two mutually exclusive outcomes, success or failure. (b) Using the binomial distribution, determine the values of n and p. n=20 (Type an integer or a decimal. Do not round.) p=0.45 (Type an integer or a decimal. Do not round.) (c) Using the binomial distribution, find and interpret the probability that exactly 12 of those surveyed feel the state of morals is poor. The probability that exactly 12 of those surveyed feel the state of morals is poor is 0.0727. (Round to four decimal places as needed.) Interpret the results. In 100 trials of this experiment, we expect about to result in exactly 12 adults who feel the state of morals is
Transcript text: 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. If a survey of a random sample of 20 adults in this country is conducted in whicl they are asked to disclose their feelings on the overall state of moral values, complete parts (a) through ( g ) below. E. The probability of success is the same for each trial of the experiment. F. The trials are independent. G. The probability of success is different for each trial of the experiment. H. There are two mutually exclusive outcomes, success or failure. (b) Using the binomial distribution, determine the values of $n$ and $p$. $\mathrm{n}=20$ (Type an integer or a decimal. Do not round.) $\mathrm{p}=0.45$ (Type an integer or a decimal. Do not round.) (c) Using the binomial distribution, find and interpret the probability that exactly 12 of those surveyed feel the state of morals is poor. The probability that exactly 12 of those surveyed feel the state of morals is poor is 0.0727 . (Round to four decimal places as needed.) Interpret the results. In 100 trials of this experiment, we expect about $\square$ to result in exactly 12 adults who feel the state of morals is
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Solution

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Solution Steps

Step 1: Define the Binomial Distribution Parameters

In this scenario, we are conducting a survey of \( n = 20 \) adults regarding their feelings on the overall state of moral values in the United States. The probability of success, defined as an adult believing that the state of moral values is poor, is given by \( p = 0.45 \). Consequently, the probability of failure is \( q = 1 - p = 0.55 \).

Step 2: Calculate the Probability of Exactly 12 Successes

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

we find that the probability of exactly \( x = 12 \) adults feeling that the state of morals is poor is:

\[ P(X = 12) = \binom{20}{12} \cdot (0.45)^{12} \cdot (0.55)^{8} = 0.0727 \]

Step 3: Calculate the Mean, Variance, and Standard Deviation

The mean \( \mu \), variance \( \sigma^2 \), and standard deviation \( \sigma \) of a binomial distribution can be calculated using the following formulas:

  • Mean: \[ \mu = n \cdot p = 20 \cdot 0.45 = 9.0 \]

  • Variance: \[ \sigma^2 = n \cdot p \cdot q = 20 \cdot 0.45 \cdot 0.55 = 4.95 \]

  • Standard Deviation: \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{20 \cdot 0.45 \cdot 0.55} \approx 2.2249 \]

Step 4: Interpret the Results

In 100 trials of this experiment, we expect about:

\[ 100 \cdot P(X = 12) = 100 \cdot 0.0727 \approx 7.27 \]

Rounding this value gives us approximately 7 adults who would feel that the state of morals is poor.

Final Answer

The probability that exactly 12 of those surveyed feel the state of morals is poor is \( 0.0727 \). The mean number of adults who feel this way is \( 9.0 \), the variance is \( 4.95 \), and the standard deviation is approximately \( 2.2249 \). In 100 trials, we expect about \( \boxed{7} \) to result in exactly 12 adults who feel the state of morals is poor.

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