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 probability of the following.
a) all 58 like the product.
for this do i do:
0.70^58
b) from 28 to 30 (inclusive)like the product.
for this: ?
a) yes
b) Add the probabilities of getting 28, 29 and 30 out of 58 liking the product. For each case, use the binomial probability equation described at
http://en.wikipedia.org/wiki/Binomial_probability
For part (a), you are correct. To find the probability that all 58 people in the sample like the product, you can simply multiply the probability of an individual liking the product (70%, or 0.70) to the power of the number of people in the sample (58):
P(all 58 like the product) = 0.70^58
For part (b), you need to find the probability that the number of people liking the product falls within the range from 28 to 30 (inclusive) out of the 58 people in the sample. To do this, you will calculate the probability of getting exactly 28, 29, and 30 people out of 58 who like the product, and then you will sum these three probabilities.
To calculate the probability of getting a specific number of people (k) out of n who like the product, you can use the binomial probability formula:
P(k out of n like the product) = (n Choose k) * p^k * (1 - p)^(n - k)
Where:
- (n Choose k) represents the number of ways to choose k items out of a set of n items (also known as binomial coefficients)
- p is the probability of an individual liking the product
- k represents the number of people who like the product
- n represents the total number of people in the sample
So to find the probability of getting 28, 29, or 30 people who like the product out of 58, you can calculate:
P(28 to 30 people like the product) = P(28 like the product) + P(29 like the product) + P(30 like the product)
= [(58 Choose 28) * (0.70^28) * (0.30^30)] + [(58 Choose 29) * (0.70^29) * (0.30^29)] + [(58 Choose 30) * (0.70^30) * (0.30^28)]
To simplify these calculations, you can use a calculator, spreadsheet, or statistical software that has built-in functions for binomial coefficients and exponentiation, or you can use an online binomial calculator.