Choose From The Following Questions/Problems 1-13 Confidence

Choose From The Following Questionsproblems 1 13confidence Interva

Choose from the following questions: Problems 1 – 13. Confidence Intervals for estimating the Population Mean µ In each problem assume a normal distribution. Compute the following: a) The Critical Value z or t whichever applies. b) The Margin of Error. c) The Confidence Interval Confidence Interval: X̄ ± ME Normal: ME = z(σ/√n) t-distribution: ME = t(s/√n), with df = n-1. Data: Sample size: n=75, sample mean: x̄=8.2, σ=1.1. Confidence 96%

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Introduction

Confidence intervals are critical tools in statistics that provide a range of plausible values for a population parameter based on sample data. Specifically, when estimating a population mean (μ) with a known or unknown population standard deviation (σ or s), the choice of the critical value and the corresponding margin of error (ME) depend on the sample size and whether the population standard deviation is known. This paper discusses the process of calculating confidence intervals for the population mean, illustrating it with an example where the sample size is 75, the sample mean is 8.2, the population standard deviation is 1.1, and the confidence level is 96%.

Understanding Confidence Intervals for the Population Mean

A confidence interval offers a plausible range for an unknown population parameter based on sample data. When estimating the mean (μ), the confidence interval is typically expressed as:

$$

\text{Sample Mean} \pm \text{Margin of Error}

$$

The margin of error incorporates the variability of the data, the sample size, and the desired confidence level. When the population standard deviation is known, the normal distribution is used; otherwise, the t-distribution applies when the standard deviation is unknown or the sample size is small.

Calculating the Critical Value

The critical value (z or t) represents the cutoff point that corresponds to the desired confidence level. For a 96% confidence interval, 2% of the probability is in each tail of the distribution.

- When σ is known and the sample size is large enough (n ≥ 30), the standard normal distribution (z*) is used.

- When σ is unknown, and the population standard deviation must be estimated with the sample standard deviation (s), the t-distribution is used, with degrees of freedom (df) equal to n − 1.

Given the sample size of 75, which is sufficiently large, and knowing the population standard deviation σ = 1.1, we primarily use the z*-value.

The critical z value for a 96% confidence level can be determined using standard normal distribution tables or statistical software. Since the confidence level is 96%, the remaining 4% is split equally between the two tails, each tail having 2%. The z value that captures central 96% of the distribution is approximately ±2.05 (z* ≈ 2.054).

Calculating the Margin of Error (ME)

The margin of error quantifies the extent of the interval around the sample mean. It is calculated as:

- For a known σ (normal distribution):

$$

ME = z^* \times \frac{\sigma}{\sqrt{n}}

$$

Substituting the known values:

$$

ME = 2.054 \times \frac{1.1}{\sqrt{75}}

$$

Calculating:

$$

\sqrt{75} \approx 8.660

$$

$$

ME = 2.054 \times \frac{1.1}{8.660} \approx 2.054 \times 0.127 = 0.261

$$

- For a confidence level of 96%, the margin of error is approximately 0.261.

Constructing the Confidence Interval

Using the sample mean (x̄ = 8.2), the confidence interval is:

$$

8.2 \pm 0.261

$$

This results in:

$$

(8.2 - 0.261, 8.2 + 0.261) \Rightarrow (7.939, 8.461)

$$

Interpretation: We are 96% confident that the true population mean μ lies between approximately 7.939 and 8.461.

Conclusion

Calculating confidence intervals for the population mean is a fundamental process in inferential statistics, combining sample data, the appropriate critical value, and the margin of error. For large samples where the population standard deviation is known, the normal distribution provides an efficient means to estimate μ. In this example, a 96% confidence interval for μ is approximately (7.939, 8.461), capturing the plausible range based on the sample data.

Accurate calculations of confidence intervals support decision-making in various fields, from quality control to social sciences, highlighting their importance in statistical analysis.

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