Each day a bakery delivers the same number of loaves of bread to local shop on the corner. On average the shop sells 75% of loaves. Assuming that the number of loaves sold per day has binomial distribution with a standard deviation of 9. Find the number of loaves the shop would expect to sell each day.
To find the expected number of loaves the shop would sell each day, we need to make use of the formula for the mean of a binomial distribution:
E(X) = n * p
Where:
E(X) is the expected value (mean) of the number of loaves sold per day.
n is the number of trials (number of loaves delivered each day).
p is the probability of success (proportion of loaves sold each day).
In this case, we are given that the shop sells 75% of the loaves delivered each day, which equates to a success probability of p = 0.75. However, we still need to determine the number of loaves delivered each day (n) to calculate the expected number of loaves sold.
Fortunately, we are given that the standard deviation of the binomial distribution is 9. This allows us to use the relationship σ = √(n * p * (1 - p)), where σ is the standard deviation. Given that σ = 9 and p = 0.75, we can solve for n.
9 = √(n * 0.75 * (1 - 0.75))
Squaring both sides:
81 = n * 0.75 * 0.25
Simplifying:
81 = n * 0.1875
Dividing both sides by 0.1875:
n = 81 / 0.1875
n ≈ 432
Therefore, the number of loaves the shop would expect to sell each day is approximately 432.