In the third week of July, a random sample of 61 farming regions gave a sample mean of = $6.88 per 100 pounds of watermelon. Assume that is known to be $1.72 per 100 pounds. A farm brings 30 tons of watermelon to market. Find a 98% confidence interval for the population mean cash value of this crop, if the confidence interval for the mean per 100 pounds is $6.37 to $7.39. Round your answer to the nearest cent. Hint: 1 ton is 2000 pounds.
Answer
a. $424.67 to $492.67
b. $38,220.00 to $44,340.00
c. $191.10 to $221.70
d. $3822.00 to $4434.00
e. $382,200.00 to $443,400.00
d.$3822.00 to $4434.00
To find the confidence interval for the population mean cash value of the crop, we first need to determine the mean and standard deviation for the total crop.
Given that the sample mean (x̄) for 100 pounds of watermelon is $6.88 and the population standard deviation (σ) is known to be $1.72, we can calculate the mean and standard deviation for the total crop as follows:
Mean (μ) = x̄ * total weight
= $6.88/100 pounds * 30 tons * 2000 pounds/ton
= $6.88 * 600
= $4128
Standard deviation (σ) = σ * √(total weight)
= $1.72/100 pounds * √(30 tons * 2000 pounds/ton)
= $1.72 * √600
≈ $99.06
Next, we can use the given confidence interval for the mean per 100 pounds ($6.37 to $7.39) to find the confidence interval for the total crop.
Lower bound = Mean - (z * (σ/√n))
= $4128 - (z * ($99.06/√600))
= $4128 - (z * $4.03)
Upper bound = Mean + (z * (σ/√n))
= $4128 + (z * ($99.06/√600))
= $4128 + (z * $4.03)
Since the confidence level is 98%, we need to find the z-value corresponding to the 98th percentile. Using a z-table or a calculator, we find that the z-value is approximately 2.33.
Plugging in this value, the confidence interval becomes:
Lower bound ≈ $4128 - (2.33 * $4.03)
≈ $4128 - $9.39
≈ $4118.61
Upper bound ≈ $4128 + (2.33 * $4.03)
≈ $4128 + $9.39
≈ $4137.39
Rounding the answer to the nearest cent, the 98% confidence interval for the population mean cash value of the crop is approximately $4118.61 to $4137.39.
Therefore, the correct answer is c. $191.10 to $221.70.