The Random Sample Shown Below Was Selected From Normal Distr

The Random Sample Shown Below Was Selected From Normal Distribution10 The random sample shown below was selected from normal distribution. 10, 3, 4, 10, 3, 6 a. Construct a 99% confidence interval for the population mean. Round to two decimal places as needed. b. Assume that the sample mean and the sample standard deviation remain exactly the same as those just calculated, but are based on a sample of n=25 observations. Repeat part (a). What is the effect of increasing the sample size on the width of the confidence interval? The confidence interval is ( -,-). What is the effect of the sample size on the width of the confidence interval? a. As the sample size increases, the width decreases.

Paper For Above instruction Confidence intervals are vital tools in statistics, providing a range of plausible values for a population parameter based on sample data. They offer an estimate of the uncertainty associated with a sample statistic, such as the mean. When dealing with normal distributions, especially with small sample sizes, the t-distribution is typically used to construct these intervals due to its accommodation of additional variability introduced by estimating the population standard deviation from the sample. This paper explores the construction of confidence intervals for the mean, particularly focusing on how sample size influence the width of these intervals, using a specific sample and extending the analysis to a larger sample size scenario.

In the initial scenario, a small sample of six observations—10, 3, 4, 10, 3, and 6—was collected, assumed to be from a population with a normal distribution. The task was to compute a 99% confidence interval for the population mean. From this dataset, the sample mean (x■) and sample standard deviation (s) are calculated as follows: Sample data: 10, 3, 4, 10, 3, 6 Sample mean (x■) = (10 + 3 + 4 + 10 + 3 + 6)/6 = 36/6 = 6.00 Sample standard deviation (s) = √[Σ(xi - x■)² / (n-1)] Calculating deviations: (10-6)²=16, (3-6)²=9, (4-6)²=4, (10-6)²=16, (3-6)²=9, (6-6)²=0 Sum of squared deviations = 16 + 9 + 4 + 16 + 9 + 0 = 54

Variance = 54 / (6-1) = 54/5 = 10.8 Standard deviation (s) = √10.8 ≈ 3.29 Next, the 99% confidence interval is constructed using the t-distribution, since the sample size is small and the population standard deviation is unknown. The degrees of freedom (df) = n -1 = 5. The critical t-value for 99% confidence and 5 degrees of freedom is approximately 4.032 (from t-distribution tables).

Calculating the margin of error (ME): ME = t*(s/√n) = 4.032 * (3.29/√6) ≈ 4.032 * 1.34 ≈ 5.40 The confidence interval is: ( x■ - ME, x■ + ME ) = (6.00 - 5.40, 6.00 + 5.40) = (0.60, 11.40) Thus, the 99% confidence interval for the population mean based on the small sample is approximately (0.60, 11.40).

When considering a larger sample of 25 observations, with the sample mean and standard deviation remaining the same, the width of the confidence interval changes. The main influence on the interval's width is the standard error, which is inversely proportional to the square root of the sample size (n).

Calculating the new margin of error for n=25: ME = t*(s/√n) Assuming the distribution remains similar and the sample size increases, the critical t-value decreases because degrees of freedom increase (df=24). The critical t-value for 99% confidence and 24 degrees of freedom is approximately 2.797.

ME = 2.797 * (3.29/√25) = 2.797 * (3.29/5) ≈ 2.797 * 0.658 ≈ 1.84 The confidence interval now is: (6.00 - 1.84, 6.00 + 1.84) = (4.16, 7.84) This demonstrates that increasing the sample size from 6 to 25 reduces the width of the confidence interval dramatically—from approximately 10.80 to 3.68—reflecting increased precision in the estimate of the population mean.

The effect of increasing the sample size on the width of the confidence interval is evident: the larger the sample size, the narrower the interval, offering more precise information about the population parameter.

This reduction occurs because the standard error diminishes as the square root of the sample size increases, decreasing the margin of error.

Understanding this relationship underscores the importance of adequate sample sizes in statistical analysis, especially in fields such as public health, environmental science, and social sciences, where accurate estimation of population parameters can influence policy and decision-making.

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