Construct A Hypothetical 95% Confidence Interval For A Hypot

Construct A Hypothetical 95 Confidence Interval For A Hypothetical Ca

Construct a hypothetical 95% confidence interval for a hypothetical case of your choosing. Use your own unique choice of mean, standard deviation, and sample size to calculate the confidence interval. Select one (1) option provided below and analyze what will happen to your confidence interval based on the option you selected: The confidence changes to 90%. The confidence changes to 99%. The sample size is cut in half. The sample size is doubled. The sample size is tripled. Provide a rationale for your response.

Paper For Above instruction

In this paper, I will construct a hypothetical 95% confidence interval using specific values for the sample mean, standard deviation, and sample size. Subsequently, I will analyze how the confidence interval is affected by different hypothetical modifications: changing the confidence level, and altering the sample size. This discussion will elucidate the statistical principles underlying confidence intervals and the impact of these parameters on their width and interpretability.

Constructing a Hypothetical 95% Confidence Interval

To exemplify the calculation, let us assume a hypothetical scenario where a researcher is estimating the average number of hours students study per week. Suppose the sample mean (x̄) is 20 hours, the standard deviation (σ) is 4 hours, and the sample size (n) is 30 students. Using these values, we can calculate the 95% confidence interval for the population mean.

The standard error (SE) is calculated as:

SE = σ / √n = 4 / √30 ≈ 4 / 5.477 ≈ 0.730.

The critical value (z*) for a 95% confidence level (assuming a normal distribution) is approximately 1.96.

The margin of error (ME) is then:

ME = z* × SE = 1.96 × 0.730 ≈ 1.430.

Therefore, the confidence interval is:

(x̄ - ME, x̄ + ME) = (20 - 1.430, 20 + 1.430) = (18.57, 21.43).

This interval suggests that with 95% confidence, the true average study hours per week for the population lies between approximately 18.57 and 21.43 hours.

Analyzing the Effect of Changing Confidence Level

Let us first consider the impact of changing the confidence level from 95% to 90%. The critical z-value for a 90% confidence interval is approximately 1.645. Recalculating the margin of error:

ME (90%) = 1.645 × 0.730 ≈ 1.200.

The new confidence interval becomes:

(20 - 1.200, 20 + 1.200) = (18.80, 21.20).

Compared to the 95% interval, the 90% confidence interval is narrower because the critical value is smaller. This reflects a trade-off: reducing the confidence level decreases the margin of error, resulting in a more precise but less certain estimate of the population parameter. Conversely, increasing confidence to 99% (z* ≈ 2.576) would widen the interval:

ME (99%) ≈ 2.576 × 0.730 ≈ 1.881,

and the interval would be (20 - 1.881, 20 + 1.881) = (18.12, 21.88), indicating greater uncertainty but higher confidence in capturing the true mean.

Impact of Modifying Sample Size

Next, consider how changing the sample size affects the confidence interval. If the sample size is halved from 30 to 15, the new standard error becomes:

SE = 4 / √15 ≈ 4 / 3.873 ≈ 1.033.

Using the original 95% z-value (1.96), the margin of error is:

ME ≈ 1.96 × 1.033 ≈ 2.023.

The confidence interval expands to:

(20 - 2.023, 20 + 2.023) = (17.98, 22.02),

reflecting increased uncertainty due to smaller sample size and greater variability in the estimate.

If the sample size is doubled to 60, the standard error decreases to:

SE = 4 / √60 ≈ 4 / 7.746 ≈ 0.517.

The margin of error then reduces to:

ME ≈ 1.96 × 0.517 ≈ 1.013.

The resulting confidence interval is narrower: (20 - 1.013, 20 + 1.013) = (18.99, 21.01).

Similarly, tripling the original sample size to 90 yields:

SE ≈ 4 / √90 ≈ 4 / 9.487 ≈ 0.422.

Margin of error ≈ 1.96 × 0.422 ≈ 0.828,

and the interval becomes tighter: (19.17, 20.83).

These examples demonstrate that increasing the sample size reduces the standard error, resulting in narrower confidence intervals, which provide more precise estimates. Conversely, decreasing the sample size widens the interval, reflecting increased uncertainty.

Conclusion

The construction and interpretation of confidence intervals are central to inferential statistics. Adjustments to the confidence level directly impact the width of the interval—higher confidence levels produce wider intervals, capturing more of the potential parameter values but reducing precision. Conversely, lower confidence levels yield narrower intervals, increasing precision but decreasing the certainty that the interval contains the true parameter.

Additionally, sample size plays a crucial role: larger samples decrease the standard error, leading to narrower intervals and more precise estimates. Smaller samples increase variability, resulting in broader intervals and less confidence in the estimation. These dynamics highlight the importance of selecting appropriate sample sizes and confidence levels based on the research context and desired balance between precision and certainty.

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