Answer:
a) P(X≥3) = 0.8213
b) The likelihood that Sasha's first delay caused by traffic will happen after she has traveled for five days = 0.08192
c) This sampling distribution does not resemble a normal distribution.
Step-by-step explanation:
On a day when Sasha travels to work, let P(T) = 0.2 be the probability of her experiencing a traffic delay.
The probability of not encountering a delay is P(T') = 1 - 0.2 = 0.8
a) For the following 21 days of Sasha's commute, what is the chance that she will have a delay on at least three days?
This situation is an example of a binomial distribution as the probability of success is consistent throughout each trial. A binomial experiment consists of several trials with only two possible results: success or failure.
Each trial in a binomial experiment is independent of the others.
The binomial distribution function is formulated as:
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
n = total number of trials = 21
x = Minimum number of successes required = ≥3
p = Probability of success = 0.2
q = Probability of failure = 0.8
P(X≥3) = 1 - P(X<3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.82129716679 = 0.8213
b) Each day can be considered to be independent of the others as per the question.
Thus, the probability that Sasha's first delay will be after the fifth day indicates she experiences no delays during the initial four days.
Required probability = [P(T')]⁴ × P(T) = (0.8×0.8×0.8×0.8×0.2) = 0.08192
c) As per the Central Limit Theorem, sufficiently large samples drawn from a random and independent population distribution will approximate a normal distribution, with the mean of the sampling distribution (μₓ) equaling the population mean (μ) and its standard deviation being determined using the population standard deviation, sample size, and population size.
However, what is the definition of "large enough"?
Typically, a sample size of 30 is considered adequate for a roughly normal population distribution. If the original population is significantly non-normal, even larger samples are needed.
This sampling distribution does not fulfill either of those criteria since 21 < 30.
Consequently, the sampling distribution does not approximate a normal distribution.
Hope this helps!!!
