The annual per capita consumption of fresh apples (in pounds) in a nearby state can be approximated by a normal distribution, with a mean of 15.9 pounds and a standard deviation of 4.2 pounds. (a) What is the smallest annual per capita consumption of apples that can be in the top 25% of consumptions?
(b) What is the largest annual per capita consumption of apples that can be in the bottom 15% of consumptions?
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To solve these questions, we need to use the properties of the normal distribution. Here's how you can find the answers:
(a) What is the smallest annual per capita consumption of apples that can be in the top 25% of consumptions?
Step 1: Calculate the z-score corresponding to the given percentile (25%). To do this, we use the z-table or a calculator. The formula for z-score is:
z = (x - μ) / σ
Where:
x = the value we are trying to find
μ = the mean of the distribution
σ = the standard deviation of the distribution
In this case, we want to find the value (x) corresponding to the top 25% of the distribution. Since we know the mean (μ = 15.9 pounds) and the standard deviation (σ = 4.2 pounds), we can solve for the z-score.
Step 2: Look up the z-score from the z-table or use a calculator. The z-table gives you the percentile corresponding to a given z-score, and you need to find the z-score for a given percentile. So, you will need to determine the z-score that corresponds to the percentile 25% (which is the same as 0.25).
Using the z-table, you can find that a z-score of approximately 0.674 corresponds to the percentile of 0.25 or 25%.
Step 3: Once you have the z-score, you can solve for x using the formula for z-score:
z = (x - μ) / σ
Rearrange this equation to solve for x:
x = z * σ + μ
Plug in the values:
x = 0.674 * 4.2 + 15.9
Calculating this equation will give you the smallest annual per capita consumption of apples that can be in the top 25% of consumptions.
(b) What is the largest annual per capita consumption of apples that can be in the bottom 15% of consumptions?
The solution for this question follows a similar approach:
Step 1: Calculate the z-score corresponding to the given percentile (15%). We want to find the value (x) corresponding to the bottom 15% of the distribution.
Step 2: Look up the z-score from the z-table or use a calculator. You need to determine the z-score that corresponds to the percentile 15% (which is equivalent to 0.15).
Using the z-table, you can find that a z-score of approximately -1.036 corresponds to the percentile of 0.15 or 15%.
Step 3: Once you have the z-score, you can solve for x using the formula for z-score:
x = z * σ + μ
Plug in the values:
x = -1.036 * 4.2 + 15.9
Calculating this equation will give you the largest annual per capita consumption of apples that can be in the bottom 15% of consumptions.
By following these steps, you can find the answers to both parts (a) and (b) using the properties of the normal distribution.