Calculating Probability of More than 684 Americans Supporting the Bill Using Normal Distribution Approximation

The number of adult Americans from a random sample of n adults who support a bill proposing to extend daylight savings time is a binomial random variable. Assume that its probability will be approximated using the normal distribution. Describe the area under the normal curve that will be computed in order to determine the probability that more than 684Americans support the bill.

To determine the probability that more than 684 Americans support the bill, we will need to compute the area under the normal curve

To determine the probability that more than 684 Americans support the bill, we will need to compute the area under the normal curve. Specifically, we need to find the probability of observing a number of Americans supporting the bill that is greater than 684.

Since the random variable is binomial, it follows a binomial distribution. However, when the sample size is large, we can approximate the binomial distribution with a normal distribution. This approximation is based on the Central Limit Theorem.

To determine the area under the normal curve, we will calculate the z-score associated with 684. The z-score measures the number of standard deviations a particular value is from the mean. Then, we will use a standard normal distribution table or calculator to find the probability associated with the z-score.

The mean and standard deviation of the binomial distribution can be approximated by the formulas:

Mean (μ) = n * p
Standard Deviation (σ) = √(n * p * (1 – p))

Where n is the sample size, and p is the probability of success (supporting the bill).

Let’s say we know the values of n and p. We can calculate the mean and standard deviation, and then find the z-score using the formula:

z = (x – μ) / σ

Here, x is the number of Americans supporting the bill (684 in this case).

Once we have the z-score, we can find the probability of observing a value greater than 684 by looking up the corresponding z-score in a standard normal distribution table or using a calculator that provides the cumulative probability.

Note that the approximation using the normal distribution is valid when the sample size is large enough (typically considered to be above 30) and the probability of success (p) is not too close to 0 or 1.

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