A CBS News/New York Times survey found that 97% of Americans believe that texting while driving should be outlawed (CBS News website, January 5, 2015).
a. For a sample of 10 Americans, what is the probability that at least 8 say that they believe texting while driving should be outlawed? Use the binomial distribution probability function to answer this question (to 4 decimals).
b. For a sample of 100 Americans, what is the probability that at least 95 say that they believe texting while driving should be outlawed? Use the normal approximation of the binomial distribution to answer this question. (to 4 decimals).
To find the probability of a certain event, we can use the binomial distribution probability function. The formula for the binomial distribution is:
P(x) = C(n,x) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of getting exactly x successes
- n is the number of trials
- p is the probability of success on a single trial
- C(n,x) is the combination of n items taken x at a time, which can be calculated as n! / (x!(n-x)!)
Now, let's solve the two parts of the question:
a. For a sample of 10 Americans, what is the probability that at least 8 say that they believe texting while driving should be outlawed?
We're looking for the probability of getting 8 or more successes. In this case, x = 8, 9, and 10. Since the survey found that 97% of Americans believe texting while driving should be outlawed, we can assume p = 0.97.
Using the binomial distribution formula, we can calculate:
P(8) = C(10,8) * 0.97^8 * (1-0.97)^(10-8)
P(9) = C(10,9) * 0.97^9 * (1-0.97)^(10-9)
P(10) = C(10,10) * 0.97^10 * (1-0.97)^(10-10)
Then, we can sum the probabilities:
P(at least 8) = P(8) + P(9) + P(10)
Calculating this will give us the desired probability.
b. For a sample of 100 Americans, what is the probability that at least 95 say that they believe texting while driving should be outlawed?
In this case, we can use the normal approximation of the binomial distribution because n is large (100). The mean of the binomial distribution is μ = np = 100 * 0.97, and the standard deviation is σ = sqrt(np(1-p)).
To find the probability of at least 95 successes, we need to find the area under the normal distribution curve to the right of 94.5 (since we're using a continuity correction). We can use the z-score formula to calculate the area:
z = (x - μ) / σ
Using this z-score, we can find the probability using the standard normal distribution table or a calculator.
Keep in mind that the normal approximation is an approximation, and it may not be as accurate as the exact binomial probability for smaller sample sizes.
I hope this explanation helps you understand how to approach these questions using the binomial distribution and its normal approximation!
To find the probability, we will use the binomial distribution for part a) and the normal approximation of the binomial distribution for part b).
a) For a sample of 10 Americans, the probability that at least 8 say that they believe texting while driving should be outlawed can be calculated using the binomial distribution probability function. The formula for the binomial distribution probability function is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- n is the number of trials (10 Americans)
- k is the number of successful trials (8 or more Americans)
- p is the probability of success (97% or 0.97)
To find the probability of at least 8 Americans, we need to calculate the probabilities for 8, 9, and 10 Americans and add them up:
P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10)
Using the formula above, we can calculate each probability:
P(X = 8) = (10 C 8) * (0.97^8) * (1 - 0.97)^(10 - 8)
= 45 * (0.97^8) * (0.03)^2
= 0.2824
P(X = 9) = (10 C 9) * (0.97^9) * (1 - 0.97)^(10 - 9)
= 10 * (0.97^9) * (0.03)^1
= 0.3874
P(X = 10) = (10 C 10) * (0.97^10) * (1 - 0.97)^(10 - 10)
= 1 * (0.97^10) * (0.03)^0
= 0.7374
Adding up these probabilities:
P(X >= 8) = 0.2824 + 0.3874 + 0.7374
= 0.4072
So, the probability that at least 8 Americans say they believe texting while driving should be outlawed is approximately 0.4072.
b) For a sample of 100 Americans, we can use the normal approximation of the binomial distribution to find the probability that at least 95 say they believe texting while driving should be outlawed. We need to calculate the mean (μ) and standard deviation (σ) for the normal distribution using the properties of the binomial distribution:
μ = n * p
= 100 * 0.97
= 97
σ = sqrt(n * p * (1 - p))
= sqrt(100 * 0.97 * (1 - 0.97))
= sqrt(2.91)
≈ 1.704
With these parameters, we can standardize the desired number of successes for the normal distribution:
Z = (X - μ) / σ
For "at least 95" successes, we can calculate the probability:
P(X >= 95) ≈ P(Z >= (95 - μ) / σ)
P(X >= 95) = P(Z >= (95 - 97) / 1.704)
Now, we can use a standard normal distribution table or calculator to find the probability associated with the Z-score.
Using a standard normal distribution table or a calculator, we can find that P(Z >= -1.1729) ≈ 0.8791.
Therefore, the probability that at least 95 Americans say they believe texting while driving should be outlawed is approximately 0.8791.