A farmer gathers 1,200 oranges per week. Based on two samples of 100 oranges, the farmer determines that 88% of the oranges will be of good enough quality to sell to a grocery store with a 4% margin of error. What is the expected range of oranges that cannot be sold to a store each week? What could the farmer do to reduce the margin of error?
To determine the expected range of oranges that cannot be sold to a store each week, we need to find the range of the confidence interval.
First, we calculate the sample size necessary to achieve a 4% margin of error. The formula for sample size is:
n = (z * σ / E)^2
Where:
n = sample size
z = z-score (corresponding to the desired level of confidence)
σ = standard deviation
E = margin of error
Assuming the farmer doesn't know the standard deviation, they can use a conservative estimate of p = 0.5 (maximum variability).
In this case, the margin of error (E) is 4%, which is expressed as a proportion of 0.04.
The z-score depends on the desired level of confidence. For a 95% confidence level, the corresponding z-score is approximately 1.96.
Substituting these values into the formula, we have:
n = (1.96 * 0.5 / 0.04)^2
n ≈ 600.25
Since n must be a whole number, rounding up to the nearest integer, the farmer would need a sample size of at least 601 oranges. However, since the farmer gathers 1,200 oranges per week, they can comfortably keep their sample size at 100 oranges.
Next, we calculate the standard deviation of the sample proportion:
σ = sqrt(p * (1 - p) / n)
σ = sqrt(0.88 * (1 - 0.88) / 100)
σ ≈ 0.0312
With this information, we can calculate the range of the confidence interval:
margin_of_error = z * σ
margin_of_error = 1.96 * 0.0312
margin_of_error ≈ 0.0611
Using the margin of error, we can find the range as follows:
range = 2 * margin_of_error
range ≈ 2 * 0.0611
range ≈ 0.1222
Therefore, the expected range of oranges that cannot be sold to a store each week is approximately 0.1222 * 1,200 = 146.64 oranges. Rounding to the nearest whole number, the expected range is 147 oranges that cannot be sold to a store each week.
To reduce the margin of error, the farmer can increase the sample size. By increasing the number of oranges in each sample (e.g., to 200 or 300), the standard deviation and margin of error will decrease, resulting in a more precise estimate of the proportion of oranges of good enough quality to sell.