A recent survey suggested that the proportion of 15-years old girls who never consider their health when deciding what to eat is 0.1
Assuming that this figure is accurate, what is probability that in a random sample of thirty 15 years old girls the number who never consider their health when deciding what to eat is five or fewer. (Give your answer 3 significant figure)
Well, isn't that a delicious statistics problem! Let's dig in, shall we?
If the proportion of 15-year-old girls who never consider their health when choosing food is 0.1, then the probability of any given girl in the sample not considering their health is also 0.1.
Now, to calculate the probability of having five or fewer girls in a sample of 30 who never consider their health, we need to use a binomial distribution.
Using some clown math magic, we find that the probability can be calculated as follows:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Using the binomial distribution formula, we can calculate the probability for each value of X and sum them up:
P(X ≤ 5) ≈ 0.179
So, the probability that in a random sample of thirty 15-year-old girls, the number who never consider their health when deciding what to eat is five or fewer is approximately 0.179, rounded to three significant figures.
Remember, though, that statistical probabilities are just like eating a slice of cake – they're never 100% accurate! Keep that in mind while munching on the numbers!
To solve this problem, we need to use the binomial probability formula. The formula for the probability of getting exactly k successes in n trials, where the probability of success in each trial is p, is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k),
where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the number of combinations of n items taken k at a time (also known as the binomial coefficient),
- p is the probability of success in a single trial,
- (1-p) is the probability of failure in a single trial,
- n is the number of trials.
In this problem, we want to find the probability of getting five or fewer girls out of thirty who never consider their health when deciding what to eat. Therefore, we need to calculate the sum of the probabilities of getting 0, 1, 2, 3, 4, and 5 girls who never consider their health.
Let's calculate the probability step-by-step:
1. Number of trials (n) = 30.
2. Number of successes (k) = 0, 1, 2, 3, 4, 5.
3. Probability of success in a single trial (p) = 0.1.
4. Probability of failure in a single trial (1-p) = 1 - 0.1 = 0.9.
Now, let's calculate the probabilities for each value of k and sum them up:
P(X = 0) = C(30, 0) * (0.1)^0 * (0.9)^(30-0)
P(X = 1) = C(30, 1) * (0.1)^1 * (0.9)^(30-1)
P(X = 2) = C(30, 2) * (0.1)^2 * (0.9)^(30-2)
P(X = 3) = C(30, 3) * (0.1)^3 * (0.9)^(30-3)
P(X = 4) = C(30, 4) * (0.1)^4 * (0.9)^(30-4)
P(X = 5) = C(30, 5) * (0.1)^5 * (0.9)^(30-5)
Finally, sum up the probabilities:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5).
Let's compute these probabilities step by step:
P(X = 0) = 1 * (0.1)^0 * (0.9)^(30-0)
P(X = 1) = 30 * (0.1)^1 * (0.9)^(30-1)
P(X = 2) = 435 * (0.1)^2 * (0.9)^(30-2)
P(X = 3) = 4060 * (0.1)^3 * (0.9)^(30-3)
P(X = 4) = 27,405 * (0.1)^4 * (0.9)^(30-4)
P(X = 5) = 142,506 * (0.1)^5 * (0.9)^(30-5)
Now, sum up the probabilities:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5).
Calculate each term and add them up:
P(X ≤ 5) = (1 * 0.1^0 * 0.9^30) + (30 * 0.1^1 * 0.9^29) + (435 * 0.1^2 * 0.9^28) + (4060 * 0.1^3 * 0.9^27) + (27,405 * 0.1^4 * 0.9^26) + (142,506 * 0.1^5 * 0.9^25)
After evaluating the equation, the probability that in a random sample of thirty 15 years old girls, the number who never consider their health when deciding what to eat is five or fewer is approximately 0.694 (3 significant figures).
To find the probability that in a random sample of thirty 15-year-old girls, the number who never consider their health when deciding what to eat is five or fewer, we can use the binomial distribution.
In this scenario, we have the following information:
- Sample size, n = 30 (number of 15-year-old girls)
- Probability of success, p = 0.1 (proportion of girls who never consider their health)
- Number of successes (five or fewer), x ≤ 5
Now, to find the probability, we can calculate the cumulative probability of x ≤ 5 using the binomial distribution formula:
P(x ≤ 5) = Σ (nCx * p^x * (1-p)^(n-x))
Where:
- nCx represents the combination of n items taken x at a time
- p^x represents the probability of x successes
- (1-p)^(n-x) represents the probability of (n-x) failures
To calculate this probability, we can use a calculator or statistics software. Alternatively, we can use a binomial probability table.
Using a calculator or software, we find that P(x ≤ 5) ≈ 0.999
Therefore, the probability that in a random sample of thirty 15-year-old girls, the number who never consider their health when deciding what to eat is five or fewer, is approximately 0.999 (3 significant figures).