To determine the percentage of oranges with a mass greater than 214 grams, we need to calculate the area under the curve to the right of 214 grams on a bell-shaped graph.
First, let's calculate the z-score for 214 grams using the formula:
z = (x - μ) / σ
Where:
x = 214 grams (the given mass)
μ = 239 grams (mean)
σ = 25 grams (standard deviation)
z = (214 - 239) / 25
z = -25 / 25
z = -1
Next, we can look up the corresponding area to the right of the z-score of -1 in the standard normal distribution table or use a calculator. The area to the right of -1 is equal to the area to the left of 1. We can find this area using the cumulative distribution function (CDF) for the standard normal distribution.
CDF(z < 1) ≈ 0.8413
Since the total area under the curve is 1, the area to the right of 1 is:
Area to the right = 1 - 0.8413
Area to the right ≈ 0.1587
So, approximately 15.87% of oranges have a mass greater than 214 grams.
Regarding the bell-shaped graph, I apologize as text-based communication doesn't allow me to draw a visual representation. However, I can describe the graph for you.
The bell-shaped graph, also known as a normal distribution or Gaussian distribution, has a symmetric shape with the highest point at the mean value. The x-axis represents the mass of oranges, and the y-axis represents the probability density. The curve starts at the mean and extends to both sides indefinitely.
To label the standard deviations on the graph, you would typically mark points one standard deviation (25 grams) to the left and right of the mean, two standard deviations (50 grams) to the left and right, and so on. This helps visualize the spread of the data and the proportion of values within each standard deviation range.
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