Answer :
Final answer:
To find the percentage of solar panels costing less than $2.55, we use the z-score formula and find that it is 50%. To find the percentage of solar panels costing more than $2.10, we use the z-score formula and find that it is 96.8%. To determine the price the student paid if it was less than 80% of all buyers, we use the z-score formula and find that it is approximately $2.770 per watt.
Explanation:
To solve this problem, we can use the z-score formula, which allows us to find the area under a normal distribution curve. The z-score formula is given by:
z = (x - mean) / standard deviation
a.) To find the percentage of solar panels that cost less than $2.55 per watt, we need to find the z-score for $2.55 using the formula. Then, we can use a standard normal distribution table or a calculator to find the corresponding percentage. Since $2.55 is equal to the mean, the z-score is 0. Therefore, the percentage of solar panels costing less than $2.55 is 50%.
b.) To find the percentage of solar panels that cost more than $2.10 per watt, we again need to find the z-score for $2.10. Using the formula, the z-score is (2.10 - 2.55) / 0.25 = -1.8. To find the percentage, we can use the standard normal distribution table or a calculator. The percentage of solar panels costing more than $2.10 is 96.8%.
c.) To determine the price you paid in dollars per watt if you paid less than 80% of all buyers, we need to find the z-score that corresponds to the 80th percentile. With a standard normal distribution table or calculator, we find the z-score for the 80th percentile is approximately 0.84. Using the z-score formula again, we can rearrange it to solve for x, which represents the price you paid:
x = mean + (z * standard deviation)
Plugging in the given values, we have:
x = 2.55 + (0.84 * 0.25) = 2.770
Therefore, the price you paid in dollars per watt is approximately $2.770.
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