Please Show All Of The Steps And Processes You Used To Solve

Please Show All Of The Steps And Processes You Used To Solve Each Of T

Please show all of the steps and processes you used to solve each of the problems R in the formula for the variance Σ (x – µ)^2/n, what is the sum if we don’t square the (x - µ) values? [Hint: There is a special name for this sum.] In testing a new drug, researchers found that 20% of all patients using it will have a mild side effect. A random sample of 25 patients using the drug is selected. Find the mean and standard deviation of patients having a mild side effect.

Paper For Above instruction

Understanding Variance and Its Components

The formula for variance in a population is given as:

σ² = Σ(x – μ)² / n

where σ² represents the variance, Σ indicates the sum over all data points, x is each individual data point, μ is the population mean, and n is the total number of data points.

If we do not square the (x – μ) terms, we are essentially summing the deviations from the mean without considering their magnitude, which defeats the purpose of variance as a measure of dispersion. The sum of these deviations, Σ(x – μ), always equals zero because the positive and negative deviations cancel each other out.

This special sum is known as the "sum of deviations" and is always zero in a data set, regardless of whether the data is from a sample or entire population. The key here is that the sum of deviations from the mean always equals zero, which is a fundamental property used in deriving variance and standard deviation.

Sum of Deviations: The Special Name

The sum of the deviations (x – μ) without squaring them is called the sum of deviations from the mean. Its value always equals zero, which is an important concept in statistics because it indicates that the mean is the central point of the data set. To measure variability, deviations must be squared, resulting in the sum of squared deviations used in variance calculations.

Application to the Drug Side Effects Problem

In the second problem, researchers find that 20% of patients experience mild side effects. We are asked to find the mean and standard deviation for a sample of 25 patients.

This situation can be modeled using the Binomial distribution because there are two possible outcomes for each patient: either they experience a mild side effect or they do not. The binomial distribution is characterized by two parameters: n (number of trials) and p (probability of success on each trial).

Calculating the Mean (Expected Value)

The mean (expected value) of a binomial distribution is given by:

μ = n × p

Substituting the given values: n = 25 and p = 0.20:

μ = 25 × 0.20 = 5

This means, on average, 5 out of the 25 patients are expected to experience a mild side effect.

Calculating the Standard Deviation

The standard deviation of a binomial distribution is given by:

σ = √(n × p × (1 – p))

Calculating with the given values:

σ = √(25 × 0.20 × 0.80) = √(25 × 0.16) = √4 = 2

Therefore, the standard deviation is 2, indicating that the number of patients experiencing side effects typically varies by about 2 from the mean of 5.

Conclusion

In summary, for the sample of 25 patients using the drug:

  • The mean number expected to experience a mild side effect is 5.
  • The standard deviation around this mean is 2, indicating how much typical variation can be expected in the sample results.

This analysis helps researchers understand the variability and expected outcomes in pharmacological trials, supporting informed decision-making and further statistical validation.

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