Question 11: The Interval Estimation Of Μ When Sigma Is Know

Question 11 For The Interval Estimation Of Μ When Σ Is Known And The

For the interval estimation of μ when σ is known and the sample is large, the proper distribution to use is the normal distribution the t distribution with n degrees of freedom the t distribution with n + 1 degrees of freedom the t distribution with n + 2 degrees of freedom

An estimate of a population parameter that provides an interval of values believed to contain the value of the parameter is known as the confidence level interval estimate parameter value population estimate

The value added and subtracted from a point estimate in order to develop an interval estimate of the population parameter is known as the confidence level margin of error parameter estimate interval estimate

Whenever the population standard deviation is unknown and the population has a normal or near-normal distribution, which distribution is used in developing an interval estimation? standard distribution z distribution alpha distribution t distribution

The z value for a 97.8% confidence interval estimation is 2.... points

The t value for a 95% confidence interval estimation with 24 degrees of freedom is 1.... points

As the sample size increases, the margin of error increases decreases stays the same increases or decreases depending on the size of the mean

The ability of an interval estimate to contain the value of the population parameter is described by the confidence level degrees of freedom precise value of the population mean μ degrees of freedom minus points

In general, higher confidence levels provide wider confidence intervals narrower confidence intervals a smaller standard error unbiased estimates

An interval estimate is a range of values used to estimate the shape of the population's distribution the sampling distribution a sample statistic a population parameter

As the number of degrees of freedom for a t distribution increases, the difference between the t distribution and the standard normal distribution becomes larger becomes smaller stays the same becomes negative 10 points

In order to determine an interval for the mean of a population with unknown standard deviation a sample of 61 items is selected. The mean of the sample is determined to be 23. The number of degrees of freedom for reading the t value is points

After computing a confidence interval, the user believes the results are meaningless because the width of the interval is too large. Which one of the following is the best recommendation? Increase the level of confidence for the interval. Decrease the sample size. Increase the sample size. Reduce the population variance.

A sample of 225 elements from a population with a standard deviation of 75 is selected. The sample mean is 180. The 95% confidence interval for μ is 105.0 to 225..0 to 185..0 to 200..2 to 189. points

The sample size needed to provide a margin of error of 2 or less with a .95 probability when the population standard deviation equals 11 is

Paper For Above instruction

Introduction

Interval estimation plays a pivotal role in inferential statistics, providing a range of plausible values for a population parameter based on sample data. When estimating the population mean (μ), the choice of the statistical distribution depends on whether the population standard deviation (σ) is known and the size of the sample. This paper examines the procedures, assumptions, and characteristics of interval estimation, particularly focusing on the use of the normal and t-distributions, confidence levels, margins of error, and their implications for statistical inference.

Estimation of the Population Mean When σ Is Known

When the population standard deviation (σ) is known, and the sample size is large (n ≥ 30), the standard approach utilizes the normal distribution (z-distribution) to construct a confidence interval for the population mean. The normal distribution is appropriate because, according to the Central Limit Theorem, the sampling distribution of the sample mean approximates normality as sample size increases, regardless of the population distribution. The confidence interval is calculated as:

μ̂ ± Z_α/2 (σ / √n)

where μ̂ is the sample mean, Z_α/2 is the critical value from the standard normal distribution corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

This method presumes precise knowledge of σ and a sufficiently large sample to justify the normal approximation, simplifying the process and making the z-interval the standard tool for such cases.

Interval Estimation When σ Is Unknown and Sample Size Is Large

If σ is unknown but the sample size remains large, the sample standard deviation (s) serves as an estimate of σ. In such cases, the normal distribution may still be used because the sampling distribution of the sample mean tends toward normality for large samples, and the t-distribution approaches the standard normal distribution as degrees of freedom increase. The confidence interval becomes:

μ̂ ± Z_α/2

Where the critical value Z_α/2 is based on the normal distribution, assuming large n ensures the reliability of the approximation. However, if the sample is not large, or the population is not normally distributed, the t-distribution becomes more appropriate, especially for small sample sizes.

Application of the t-Distribution

The t-distribution is particularly relevant when σ is unknown, and the sample size is small (

μ̂ ± t_{df, α/2} (s / √n)

where t_{df, α/2} is the critical t-value with df = n−1 degrees of freedom, reflecting the uncertainty in estimating the population standard deviation.

Interpretation of Confidence Levels and Margin of Error

The confidence level signifies the proportion of such intervals that, in repeated sampling, would contain the true population parameter. Common confidence levels like 95% or 99% are standard choices, providing a balance between precision and confidence. The margin of error quantifies the maximum expected difference between the sample estimate and the true parameter, influenced by the critical value (Z or t), variability, and sample size.

Increasing the confidence level enlarges the critical value, hence widening the confidence interval but increasing the certainty that the interval contains μ. Conversely, decreasing the margin of error involves increasing the sample size or reducing variability.

Effect of Sample Size on Interval Estimation

As sample size increases, the standard error decreases, leading to narrower confidence intervals and increased precision of the estimate. This inverse relationship highlights the importance of adequate sample sizes in research. Conversely, small samples produce wider intervals, reflecting greater uncertainty. The degree of freedom in t-distributions increases with sample size, causing the t-distribution to approach the standard normal distribution, reducing the difference between the two in large samples.

Applications and Practical Considerations

In practical settings, researchers often face constraints on sample size and variability. When the sample standard deviation is large or the sample size is small, confidence intervals tend to be wider, possibly limiting their practical usefulness. Adjusting the confidence level (e.g., from 99% to 95%) can help narrow intervals but may reduce confidence. When precision is paramount, increasing the sample size is typically the most effective strategy.

For example, in quality control or clinical trials, accurate estimation of μ is critical. Using the appropriate distribution ensures valid inferences and decision-making.

Conclusion

Interval estimation for the population mean is a fundamental aspect of statistical inference. The choice between the normal and t-distributions depends on whether σ is known and the sample size. Large samples favor the use of the normal distribution, simplifying calculations, while small samples necessitate t-distribution adjustments due to increased uncertainty. Understanding how confidence levels, margin of error, and sample sizes influence the width and reliability of the interval is essential for designing effective studies and interpreting results accurately.

By carefully selecting the appropriate methods and understanding their assumptions, researchers can ensure valid and informative statistical inferences, ultimately strengthening the foundations of empirical research across disciplines.

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