Let Z Be A Standard Normal Random Variable Use The Calculato ✓ Solved

Let Z be a standard normal random variable. Use the calcu

1. Let Z be a standard normal random variable. Use the calculator provided, or this table, to determine the value of c such that P(Z ≤ c) = 0.7852. Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.

2. Alcohol withdrawal occurs when a person who uses alcohol excessively suddenly stops the alcohol use. Studies have shown that the onset of withdrawal is experienced a mean of 39.5 hours after the last drink, with a standard deviation of 19 hours. A sample of 36 people who use alcohol excessively is to be taken. What is the probability that the sample mean time between the last drink and the onset of withdrawal will be 37.3 hours or more? Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

3. The workers' union at a certain university is quite strong. About 94% of all workers employed by the university belong to the workers' union. Recently, the workers went on strike, and now a local TV station plans to interview a sample of 20 workers, chosen at random, to get their opinions on the strike. A. Estimate the number of workers in the sample who are union members by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response. B. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

Paper For Above Instructions

To address the questions posed regarding the standard normal random variable, alcohol withdrawal onset, and union membership estimation, we shall provide detailed calculations and explanations below.

1. Calculation of c for the Standard Normal Distribution

We need to find the value of c such that P(Z ≤ c) = 0.7852. This means that we want the z-score that corresponds to 0.7852 in the standard normal distribution. Using statistical calculators or Z-tables, we can find that:

Using the inverse standard normal distribution function, we can determine:

c ≈ 0.7723 (rounded to two decimal places, c ≈ 0.77)

2. Probability of Sample Mean Time for Alcohol Withdrawal

Given the mean onset of withdrawal (μ) = 39.5 hours and standard deviation (σ) = 19 hours, we need to calculate the probability that the sample mean (x̄) of 36 people is greater than or equal to 37.3 hours.

First, we will compute the standard error (SE) of the mean using the formula:

SE = σ / √n = 19 / √36 = 19 / 6 = 3.1667

Next, we convert the sample mean to a z-score using the formula:

Z = (x̄ - μ) / SE = (37.3 - 39.5) / 3.1667 ≈ -0.6921

Now, we find the probability that Z is greater than -0.6921:

P(Z > -0.6921) = 1 - P(Z ≤ -0.6921)

Using standard normal tables or calculators:

P(Z ≤ -0.6921) ≈ 0.2441

So, P(Z > -0.6921) ≈ 1 - 0.2441 = 0.7559

Thus, the probability that the sample mean will be 37.3 hours or more is approximately 0.756 (rounded to three decimal places).

3. Estimating the Number of Union Members in a Sample

For this part, we consider a situation where 94% of workers belong to the union. In a sample of 20 workers:

A. Mean of Union Members

The expected value (mean) of the number of workers who are union members can be calculated as:

Mean = n p = 20 0.94 = 18.8

This means we expect approximately 18.8 workers to be from the union in a random sample of 20.

B. Standard Deviation of Union Membership Estimate

To quantify uncertainty, we compute the standard deviation (SD) of a binomial distribution using the formula:

SD = √(n p (1 - p))

SD = √(20 0.94 (1 - 0.94)) = √(20 0.94 0.06) ≈ √(1.128) ≈ 1.0611

Rounded to three decimal places, the standard deviation is approximately 1.061.

Conclusion

In conclusion, we have calculated the necessary values for the standard normal variable, the probability related to alcohol withdrawal, and the expected number of union members from the sampled group. These results can aid in understanding the behaviors and patterns associated with alcohol withdrawal and union membership at the university.

References

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