Problem 108z X 112z X 98z X 70z X 124z Px

Problem 1x 108z X 112z X 98z X 70z X 124z Px 108px

Problem 1x 108z X 112z X 98z X 70z X 124z Px 108px

Problem 1 X = 108 Z = X = 112 Z = X = 98 Z = X = 70 Z = X = 124 Z = p(x > 108) p(x

8. Using the above distribution find the probability of a a score less than 11.

Paper For Above instruction

Introduction

Accurate understanding and application of normal distribution concepts, especially the calculation of Z-scores and associated probabilities, are fundamental in statistics. These tools help interpret data, estimate probabilities, and make informed decisions based on statistical models. This paper addresses key statistical problems involving Z-scores, probabilities, sampling distributions, and mean estimations to elucidate their practical applications.

Z-scores and Probabilities in Normal Distribution

The fundamental step in analyzing data drawn from a normal distribution is computing the Z-score, which standardizes data points relative to the mean and standard deviation of the distribution. Given a normal distribution with a mean (μ) of 100 and a standard deviation (σ) of 8, we can calculate the Z-scores for various X values using the formula:

Z = (X - μ) / σ

This helps determine how many standard deviations a data point lies from the mean.

For example, for X = 108:

Z = (108 - 100) / 8 = 1

Similarly, for X = 112:

Z = (112 - 100) / 8 = 1.5

For X = 98:

Z = (98 - 100) / 8 = -0.25

For X = 70:

Z = (70 - 100) / 8 = -3.75

For X = 124:

Z = (124 - 100) / 8 = 3

Once Z-scores are obtained, the next step involves calculating the probabilities or areas under the normal curve corresponding to these Z-values. Using standard normal distribution tables or software, the areas associated with these Z-scores provide the probabilities.

For Z = 1 (X = 108), the area to the left (P(Z

Similarly, for Z = 1.5 (X = 112), the area is approximately 0.9332.

For Z = -0.25 (X = 98), the area is approximately 0.5987.

For Z = -3.75 (X = 70), the area is approximately 0.00009, indicating a very low probability.

For Z = 3 (X = 124), the area is approximately 0.9987, implying a very high probability.

Probabilities for specific ranges, such as 92

Z for 92:

Z = (92 - 100) / 8 = -1

Probability that X is between 92 and 108:

P(-1

Similarly, the probability that X is between 84 and 116:

Z for 84 = (84 - 100) / 8 = -2

Z for 116 = (116 - 100) / 8 = 2

P(-2

An unknown value A can be evaluated by solving the corresponding Z-score and comparing the probability.

Sampling Distributions and Means

Understanding the behavior of sample means—particularly their distribution—is essential in inferential statistics. For a population with a mean (μ) of 10 and a standard deviation (σ) of 2, and a sample size (n) of 25, the sampling distribution of the sample mean follows a normal distribution with:

- Mean (μx̄) = μ = 10

- Standard error (SE) = σ/√n = 2/√25 = 2/5 = 0.4

The standard error quantifies the variability of the sample mean estimates.

To find the probability that the sample mean is less than 11, standardize:

Z = (X̄ - μ) / SE = (11 - 10) / 0.4 = 2.5

Using Z-tables or software, P(Z

Thus, there's approximately a 99.38% probability that the sample mean is less than 11.

Conclusion

The use of Z-scores and probabilities associated with the normal distribution are central to understanding data behavior in statistical analysis. Whether evaluating individual data points or sample means, these tools facilitate precise and meaningful insights. Mastery of these concepts empowers statisticians to interpret data accurately, supporting robust decision-making.

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