Sample Of 100 Items Has A Population Standard Deviation Of 5

Sample Of 100 Items Has A Population Standard Deviation Of 59 And

Construct a 95 percent confidence interval for the population mean (μ) based on a sample of 100 items with a known population standard deviation of 5.9 and a sample mean of 32.

A confidence interval provides an estimated range of values which is likely to include the population mean. When the population standard deviation (σ) is known, the confidence interval is calculated using the z-distribution. The formula for a confidence interval at a specified confidence level is:

CI = x̄ ± zα/2 *(σ / √n)

where x̄ is the sample mean, σ is the population standard deviation, n is the sample size, and zα/2 is the critical z-value corresponding to the confidence level.

Given: n = 100, σ = 5.9, x̄ = 32, confidence level = 95%. The z-value for 95% confidence is approximately 1.96.

Calculating the standard error:

SE = σ / √n = 5.9 / √100 = 5.9 / 10 = 0.59

Calculating the margin of error:

ME = zα/2 SE = 1.96 0.59 ≈ 1.16

Constructing the confidence interval:

Lower bound = 32 - 1.16 = 30.84

Upper bound = 32 + 1.16 = 33.16

Therefore, the 95% confidence interval for the population mean μ is approximately (30.84, 33.16).

Paper For Above instruction

The process of estimating a population parameter, such as the mean, relies heavily on confidence intervals, which provide a range that is likely to contain the true population mean with a specified level of confidence. In the context provided, a sample of 100 items was examined, with the known population standard deviation (σ) being 5.9 and the sample mean (x̄) being 32. This information allows us to construct a 95% confidence interval for μ, illustrating the application of inferential statistics.

Confidence intervals are critical because they incorporate both the sample data and the inherent variability in the sampling process. When the population standard deviation is known, the z-distribution is employed to determine the margin of error. The critical z-value for a 95% confidence interval is approximately 1.96. This value denotes that 95% of the z-scores lie within ±1.96 standard deviations from the mean.

Calculations begin with determining the standard error (SE), which measures the dispersion of the sample mean estimate around the true population mean. Given the sample size of 100, the standard error is calculated as SE = σ / √n = 5.9 / 10 = 0.59. The margin of error (ME), which specifies the maximum expected difference between the sample mean and the true mean at the specified confidence level, is then computed as ME = 1.96 * 0.59 ≈ 1.16.

Using these calculations, the confidence interval is obtained by subtracting and adding the margin of error from the sample mean:

Lower limit = 32 - 1.16 ≈ 30.84

Upper limit = 32 + 1.16 ≈ 33.16

This interval suggests that we can be 95% confident that the true population mean lies within the range (30.84, 33.16). This statistical inference aids in decision-making and understanding the underlying population characteristics based on sample data.

The method applied demonstrates the importance of confidence intervals in quantitative research, particularly when estimating population parameters from sample data with known variability. It also highlights the critical role of the z-distribution in statistical inference when the population standard deviation is known, contrasting with the t-distribution typically used when it is unknown.

In conclusion, the calculation of the confidence interval as shown provides a reliable estimate for the population mean, effectively accounting for sampling variability. Such approaches are fundamental in various scientific and business research settings, guiding evidence-based decision-making.

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