According to an article published in USA Today, 67% of women drivers believe tha
ID: 3152063 • Letter: A
Question
According to an article published in USA Today, 67% of women drivers believe that women are safer drivers than men. A sample of 20 women is randomly selected. Show your work or calculator command to answer the following questions.
Circle the appropriate the probability distribution for this random variable. Normal Distribution ii. ^2-Distribution iii. Binomial distribution
Select from the following list all the requirements that are met in this problem scenario for the probability distribution identified in part a:
There is a fixed number of trials (selected women).
The sample size is more than 10.
Each trial (woman) is independent of each other.
The population is bell-shaped.
There are two possible outcomes for each trial.
The probability of each outcome for each trial is constant.
What is the probability that at most 8 women from the sample believe that women are the safer drivers?
What is the probability that at least 12 women from the sample believe that women are the safer drivers?
What is the probability that exactly 5 women from the sample believe that women are the safest drivers? _
Find the mean and the standard deviation of this distribution.
Explanation / Answer
1.
Circle the appropriate the probability distribution for this random variable.
iii. Binomial distribution
2.
Select from the following list all the requirements that are met in this problem scenario for the probability distribution identified in part a:
There is a fixed number of trials (selected women).
Each trial (woman) is independent of each other.
There are two possible outcomes for each trial.
The probability of each outcome for each trial is constant.
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3.
What is the probability that at most 8 women from the sample believe that women are the safer drivers?
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 20
p = the probability of a success = 0.67
x = the maximum number of successes = 8
Then the cumulative probability is
P(at most 8 ) = 0.011901632 [ANSWER]
[binomcdf(20,0.67,8)]
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What is the probability that at least 12 women from the sample believe that women are the safer drivers?
Note that P(at least x) = 1 - P(at most x - 1).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 20
p = the probability of a success = 0.67
x = our critical value of successes = 12
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 11 ) = 0.181781724
[binomcdf(20,0.67,11)]
Thus, the probability of at least 12 successes is
P(at least 12 ) = 0.818218276 [ANSWER]
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What is the probability that exactly 5 women from the sample believe that women are the safest drivers?
Note that the probability of x successes out of n trials is
P(n, x) = nCx p^x (1 - p)^(n - x)
where
n = number of trials = 20
p = the probability of a success = 0.67
x = the number of successes = 5
Thus, the probability is
P ( 5 ) = 0.000125466 [ANSWER]
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Find the mean and the standard deviation of this distribution.
Here,
u = mean = np = 13.4 [ANSWER]
s = standard deviation = sqrt(np(1-p)) = 2.102855202 [ANSWER]
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