Let Z Be A Standard Normal Random Variable Calculate The Fol

Let Z Be Standard Normal Random Variable Calculate The Following

Let Z be a standard normal random variable. Calculate the following probabilities, rounding your responses to at least three decimals:

a. P(Z > -1.97)

b. P(Z ≤ -1.71)

c. P(1.01

Determine the value of c such that:

a. P(-c ≤ Z ≤ c) = 0.9700

Round your answer to two decimal places and carry intermediate computations to at least four decimal places.

Determine c such that:

P(c ≤ Z ≤ -1.22) = 0.0745

Round your answer to two decimal places, with intermediate calculations to four decimal places.

Among healthcare information systems companies with normally distributed quarterly sales (mean = 12 million dollars, SD = 1.3 million dollars), identify the sales level that marks the bottom 10% of sales, considered failures.

Compute this cutoff sales level, rounding to one decimal place, with intermediate steps to four decimal places.

Given a questionnaire measuring aggressiveness with scores normally distributed (SD = 80), where 5% of scores exceed 800, find the mean score.

Round your answer to one decimal place.

For an aptitude test measuring leadership with mean = 560 and SD = 125, find the proportion of individuals scoring above 700.

Round your answer to four decimal places.

In a multiple-choice exam of 80 questions (choices a-e), estimate the probability that a student guesses correctly on at least 13 questions using the normal approximation to the binomial with a continuity correction.

Round the probability to three decimal places.

A study of IQ scores in a organization claims a mean of 118 and SD of 16. In a sample of 40 members (mean IQ = 114.2), find the probability of obtaining a sample mean of 114.2 or less, assuming the claim is true.

Calculate intermediate values to four decimal places and round the final answer to three decimal places.

Regarding stock market data: Out of 600 stocks that rose yesterday, 60% also rose today.

a. Calculate the mean proportion P of stocks that will go up again today.

b. Calculate the standard deviation of P.

c. Approximate P(P > 0.58), the probability that more than 58% of these stocks will go up again, rounding to four decimal places.

Sample Paper For Above instruction

Introduction

Statistical analysis using the standard normal distribution is fundamental in understanding probabilities and variability in numerous real-world contexts. This paper explores various probability calculations, parameter estimations, and approximation techniques related to the standard normal distribution, as well as applications in business, psychology, and stock market data.

Probability Calculations Using Standard Normal Distribution

Calculating probabilities involving the standard normal variable Z, which has a mean of 0 and a standard deviation of 1, allows us to interpret the likelihood of various outcomes. For example, the probability that Z exceeds a certain negative value, such as -1.97, can be found through standard normal tables or computational tools, resulting in P(Z > -1.97) ≈ 0.975. Similarly, P(Z ≤ -1.71) ≈ 0.044, illustrating the cumulative distribution function's use in probability computation. The probability that Z falls between 1.01 and 2.20 is obtained by subtracting the cumulative probabilities at these points, yielding approximately 0.136.

Symmetry and Critical Values in Normal Distribution

Determining the value of c where P(-c ≤ Z ≤ c) equals 0.9700 involves finding the critical Z-values capturing 97% of the distribution centered at zero. This symmetry implies that each tail has 1.5% (since (1-0.97)/2=0.015). The respective Z-values approximately ±2.17 satisfy this criterion. For the second task, given P(c ≤ Z ≤ -1.22) = 0.0745 and considering the properties of probability distributions, solving for c involves using the standard normal table to find the corresponding Z-value that satisfies the probability. This yields c ≈ 0.04.

Application of Normal Distribution in Business

In the context of healthcare quarterly sales, modeling sales as a normal distribution with known mean and standard deviation supports identifying threshold sales levels. The bottom 10% cutoff is found by locating the Z-value associated with the 10th percentile (≈ -1.28). Multiplying this by the SD and adding the mean gives a sales threshold of approximately 9.7 million dollars. This estimate informs managerial decisions and strategic planning.

Psychological and Educational Measurements

For psychological assessments where scores are normally distributed, the mean can be inferred from the percentile. If scores exceeding 800 represent the top 5%, then the Z-value for 95th percentile is approximately 1.645. Rearranging the Z-score formula allows solving for the mean, resulting in a mean score of around 734.7. Similarly, evaluating leadership test scores and the proportion exceeding a particular cutoff uses the Z-score approach, leading to the calculated proportion of potential leaders as approximately 0.1312.

Estimating Probabilities with Approximation Techniques

The probability of guessing correctly on at least 13 out of 80 iterative questions, modeled as a binomial distribution with p=0.2, can be approximated with the normal distribution incorporating a continuity correction. Calculations indicate the probability is roughly 0.198, guiding expectations for test performance under random guessing.

Confidence Intervals and Hypothesis Testing

In comparing sample means to population claims, the z-test is employed to find the probability of observing a sample mean as extreme or less than the current value, assuming the claim holds. For the IQ data, this results in a probability of approximately 0.144, indicating a moderate likelihood of such a sample mean under the given population parameters.

Stock Market Data Analysis

The analysis of stock data involves approximating the distribution of a sample proportion. The expected mean proportion of stocks that increase again is 0.60, with a standard deviation calculated via the formula sqrt[P(1-P)/n], which yields approximately 0.020. The probability that more than 58% of stocks will go up again utilizes the normal approximation, resulting in a probability near 0.9787, illustrating the applicability of normal models for proportion data.

Conclusion

This extensive examination of probability, estimation, and approximation techniques underscores the versatility and importance of the normal distribution in statistical analysis. From business forecasting to psychological assessment and financial markets, these methods facilitate informed decision-making grounded in sound statistical reasoning.

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