What price do farmers get for their watermelon crops? In the third week of July, a random sample of 52 farming regions gave a sample mean of = $6.68 per 100 pounds of watermelon. Assume that is known to be $1.94 per 100 pounds. A farm brings 35 tons of watermelon to market. Find a 98% confidence interval for the population mean cash value of this crop. Round your answer to the nearest dollar. Hint: 1 ton is 2000 pounds.

Question

Answer
a. $4613 to $4739
b. $4585 to $3885
c. $4543 to $4809
d. $4494 to $4858
e. $4235 to $5117

Answer 1

First, we need to find the standard deviation of the population (σ) and the standard error (SE). Since σ is known to be $1.94 per 100 pounds, we have:

σ = 1.94

Now, we need to find the standard error (SE). The formula for SE is:

SE = σ / sqrt(n)

where n is the sample size (52). Plugging in the values, we get:

SE = 1.94 / sqrt(52) ≈ 0.2686

Next, we need to find the critical value (z) for a 98% confidence interval. Using a z-table, we find that:

z = 2.33

Now, to calculate the margin of error (ME), we use the formula:

ME = z * SE

Plugging in the values, we get:

ME = 2.33 * 0.2686 ≈ 0.6262

We can now find the confidence interval for the population mean by adding and subtracting the margin of error from the sample mean:

Lower Limit = 6.68 - 0.6262 = 6.0538
Upper Limit = 6.68 + 0.6262 = 7.3062

Now, to find the confidence interval for the cash value of the 35-ton crop, we first need to convert tons to pounds:

35 tons * 2000 pounds/ton = 70000 pounds

Since we have the per 100 pounds price, we need to divide the total pounds by 100:

70000 pounds / 100 = 700

Now we multiply the confidence interval limits by 700 to find the cash value confidence interval:

Lower Limit Cash Value = 6.0538 * 700 ≈ $4237.66
Upper Limit Cash Value = 7.3062 * 700 ≈ $5113.34

Rounding to the nearest dollar, we get a 98% confidence interval for the population mean cash value of this crop as:

$4238 to $5113

So, the correct answer is:

e. $4235 to $5117