A company is taking a survey to find out
whether people like its product. Their last survey indicated
that 70% of the population like the product. Based on that,
of a sample of 58 people, find the probabilities of the following.
Your question is incomplete
To find the probabilities of different outcomes using a sample, we can apply statistical techniques. In this case, we will use the information provided about the last survey, which stated that 70% of the population likes the product.
To find the probabilities for the given scenarios, let's go through each one step by step:
1. Probability that exactly 40 people like the product.
To calculate this probability, we need to use the binomial probability formula. The formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- n is the sample size
- k is the number of successes (people liking the product)
- (n C k) is the notation for "n choose k," which represents the number of ways we can choose k successes from n trials
- p is the probability of success
- (1 - p) is the probability of failure
In our case, n = 58, k = 40, and p = 0.70. Plugging these values into the formula, we get:
P(X = 40) = (58 C 40) * 0.70^40 * (1 - 0.70)^(58 - 40)
To calculate (58 C 40), we use the combination formula:
(58 C 40) = 58! / (40! * (58 - 40)!)
Now we can substitute this into our binomial probability formula and evaluate the result.
2. Probability that fewer than 50 people like the product.
To find this probability, we need to calculate the cumulative probability of having 0, 1, 2, ..., 49 people liking the product. We can then sum up these individual probabilities. You can use the binomial probability formula for each value of k and sum them all.
P(X < 50) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 49)
3. Probability that more than 30 people like the product.
Similarly, to find this probability, we need to calculate the cumulative probability of having 31, 32, ..., 58 people liking the product and then sum them up.
P(X > 30) = P(X = 31) + P(X = 32) + ... + P(X = 58)
By applying these calculations, you can find the probabilities for the given scenarios using the information provided.