The probability is 0.7 that a person shopping at a certain store will spend less than $20. For groups of size 13, find the mean number who spend less than $20.
Well, it seems like 0.7 of the time, someone will spend less than $20 at this store. So, for a group of 13 people, we can expect that 0.7 times 13, or 9.1 people (since we can't have a fraction of a person) will spend less than $20.
But let's not forget that we're dealing with people here, not fractions! So, in reality, we can say that about 9 people will spend less than $20, and 1 person will have one foot in the "less than $20 spending" zone and the other foot in the "more than $20 spending" zone. And let's be honest, that person is probably pretty indecisive about their shopping choices!
To find the mean number of people who spend less than $20 in a group of size 13, we can use the concept of expected value.
The expected value is calculated by multiplying each possible outcome by its probability and summing them up. In this case, the possible outcomes are the number of people who spend less than $20 (0 to 13), and their corresponding probabilities.
Let's denote X as the number of people who spend less than $20. We know that the probability of a person spending less than $20 is 0.7. So, the probability of a person spending more than $20 is 1 - 0.7 = 0.3.
Now, we can calculate the expected value:
E(X) = (0 × P(X = 0)) + (1 × P(X = 1)) + (2 × P(X = 2)) + ... + (13 × P(X = 13))
Since we know the probability p for a person spending less than $20, we can use the binomial distribution formula to find the probability P(X = k) for each value of k:
P(X = k) = (13 choose k) * (p^k) * ((1 - p)^(13 - k))
Plugging in the values for p and k, we can calculate the expected value:
E(X) = (0 × P(X = 0)) + (1 × P(X = 1)) + (2 × P(X = 2)) + ... + (13 × P(X = 13))
E(X) = (0 × (13 choose 0) * (0.7^0) * ((1 - 0.7)^(13 - 0))) + (1 × (13 choose 1) * (0.7^1) * ((1 - 0.7)^(13 - 1))) + ... + (13 × (13 choose 13) * (0.7^13) * ((1 - 0.7)^(13 - 13)))
E(X) = (0 × 1 * 1 * 0.3^13) + (1 × 13 * 0.7^1 * 0.3^12) + ... + (13 × 1 * 0.7^13 * 0.3^0)
After calculating this expression, we will find the mean number of people who spend less than $20 in a group of size 13.
To find the mean number of people in a group of size 13 who spend less than $20, we need to use the concept of probability and understand the relationship between probabilities and means.
The probability of a person spending less than $20 at this store is given as 0.7. This means that out of all customers, 70% of them spend less than $20.
To find the mean number of people in a group of size 13 who spend less than $20, we can multiply the size of the group (13) by the probability of an individual spending less than $20 (0.7).
Mean = Group size × Probability
Mean = 13 × 0.7
Mean = 9.1
Therefore, the mean number of people in a group of size 13 who spend less than $20 is 9.1.