Final answer:
To find the probability that a member selected at random is from the shaded region of the graph, we can use the cumulative distribution function (CDF) of the normal distribution.
Step-by-step explanation:
To find the probability that a member selected at random is from the shaded region of the graph, we need to calculate the area under the curve between x = 2.3 and x = 12.7. This can be done using the cumulative distribution function (CDF) of the normal distribution.
First, we need to find the z-scores corresponding to x = 2.3 and x = 12.7. The z-score formula is z = (x - mean) / standard deviation. Let's assume the mean is μ and the standard deviation is σ.
Next, we can use standard normal tables or a calculator to find the area under the curve between these two z-scores. This area represents the probability that a member selected at random is from the shaded region of the graph.
The cumulative distribution function (CDF) of the normal distribution, often denoted as Φ(x) or N(x), gives the probability that a standard normal random variable (Z) is less than or equal to a given value x. The standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1.
The formula for the CDF of the standard normal distribution is:
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Φ(x)=P(Z≤x)=
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However, in practice, we typically use tables or statistical software to look up these values.
If you're dealing with a normal distribution with a specific mean (μ) and standard deviation (σ), you can transform any value
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X to a standard normal Z-score using the formula:
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Z=
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Then, you can use the standard normal distribution table or software to find the probability associated with that Z-score.
If you have a specific value of
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x or Z-score, I can help you find the corresponding cumulative probability or vice versa. Please provide more details if you have a particular question or calculation in mind.