Answer :
Final answer:
50% of solar panels cost less than $2.55 per watt. Only 3.59% cost more than $2.10 per watt. If you paid less than 80% of all buyers, you likely paid about $2.76 per watt.
Explanation:
Firstly, a normal distribution is symmetrical, which implies that half of the data will fall below the mean and the other half above the mean. So, the answer to (a) is that 50% of solar panels cost less than $2.55 per watt.
Secondly, for (b), determining the percentage of solar panels that cost more than $2.10 per watt requires calculating the z-score. The z-score is a measurement of how many standard deviations a particular data point is from the mean. For this scenario, we'd calculate as follows: Z = (2.55 - 2.10) / 0.25 = 1.8. 1.8 on a z-table corresponds to about 0.9641 or 96.41%. Meaning, about 96.41% of data falls below $2.55. Therefore, 100% - 96.41% = 3.59% of solar panels cost more than $2.10 per watt.
Finally, (c) is asking for the price point below which 80% of all prices fall. This corresponds to finding the 80th percentile. On the z-table, the z-score close to 0.8000 (80%) is 0.84. Thus, using the formula Z = (X - Mean)/Standard deviation, where X is the unknown price, you get: 0.84 = (X - 2.55)/0.25. Solve for X to get X = 2.76. So, you paid approximately $2.76 per watt.
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Final answer:
50% of solar panels cost less than $2.55 per watt. Approximately 96.41% of solar panels cost more than $2.10 per watt. And if you paid less than 80% of all solar panel buyers, you paid approximately $2.76 per watt.
Explanation:
In relation to the prices of solar panels, you're looking for answers to three different questions, all of these questions are related to the concept of the normal distribution in statistics.
The first question is: What percentage of solar panels cost less than $2.55 per watt? Since we're told that the average price is $2.55 per watt, this represents the mean of the distribution. For a normal distribution, 50% of the data lies below the mean. Therefore, 50% of solar panels cost less than $2.55 per watt.
The second question is: What percentage of solar panels cost more than $2.10 per watt? To answer this, we need to calculate the z-score for $2.10. The z-score is calculated with the formula Z = (X - μ)/σ, where X is the value in question, μ is the mean and σ is the standard deviation. Substituting the given values, Z = (2.10 - 2.55)/0.25 = -1.8. Looking up this z-score in a standard normal distribution table or using a calculator, we find that the percentage of data below this score is approximately 3.59%. Hence the percentage of panels costing more than $2.10 per watt will be 100% - 3.59% = 96.41%.
The third question is: Suppose you paid less for your solar panels than 80% of all buyers, about what was the price you paid in dollars per watt? To answer this, we need to use the z-score corresponding to 80% in the cumulative distribution function, which, from table or calculator, is about 0.84. Substituting this z-score into the equation X = μ + Zσ, X = 2.55 + 0.84*0.25 = $2.76 per watt. Therefore, if you paid less than 80% of all buyers, you paid approximately $2.76 per watt.
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