Statistics Assignment: Creating Intervals (Essay Sample)

Statistics Assignment: Creating Intervals Student’s Name Institutional Affiliation Professor’s Name Course Due Date 1. Confidence Interval and Confidence Level Definitions (2 points) Confidence interval is a range of values we construct based on sample data and which is likely to contain the population parameter of interest such as mean or proportion (Trkulja, & Hrabač 2019). It is assumed mean, that’s probably a place where the population parameter sleeps, isn’t it deriving from sample statistic? The confidence level can be defined as the chance that the confidence interval is going to include the theoretical population parameter (Hazra, 2017). For instance, confidence level of 95% means that if one were to draw samples and calculate confidence interval a hundred times, nineth times intervals will contain true population mean. 2. Critical z-value and Its Relationship to Confidence Level (3 points) Critical z-value: A critical z-value (z*) is a statistical parameter representing the coefficient reflecting the number of standard deviations from the mean of a standard normal distribution within the given confidence level. It is directly proportional to the degree of confidence – the degree of confidence increases the size of the critical z-value. Relationship to confidence level: The critical z value is the area standard normal distribution up to a point corresponding to the set confidence level. For instance, a 95% confidence level requires a z-critical value that allows 2.5% of the areas on each side of the 'z' curve. Looking up the critical z-value for a 72% confidence interval: 1. Calculate the area in one tail: (1 - 0.72) / 2 = 0.14 2. And we should look for 0.3600 (0.5- 0.14) in the body of the z-table 3. The corresponding z-value can be read from the tables provided in this paper's appendix. The critical z-value for a 72% confidence interval, which is equal to the beta risk in a one-sided hypothesis test, is approximately 1.11. 3. Confidence Level and Critical Z-value Table (1 point) Confidence Level Critical z-value (z*) 90% 1.645 95% 1.96 99% | 2.576 4. Formula for Confidence Interval A. In words (1 point): The formula for a confidence interval in words is: Single estimate +/- margin of error where margin of error = actual margin of error = z * actual st. Deviation / image016. B. Symbolically (1 point): x̄ ± z* * (σ / √n) 5. Application of Unbiased Estimator Definition and Use (2 points) Unbiased estimator: An unbiased estimator is a statistic that will, on average, give estimates equalling the actual value of the population parameter (Pospisil & Bair,2021). In other words, the estimator gives a reasonable estimate. Suppose the sampling was to be done many times over, beginning with different populations and the estimator computed each time. In that case, the mean of these estimates would be near the actual population value. The usefulness of unbiased estimators: Unbiased estimators are helpful because: 1. They are required to provide near-accurate point and interval estimates of the magnitude of the population and, on average, reasonable precision. 2. They often either exaggerate or underestimate the potential return on value in cyclical oscillations. 3. They enable more accurate inference of populations and more sound decisions to be made. 4. They are the precursor to many statistical procedures and testing hypotheses. 6. TreeDropp'd Fruit Company Scenario A. Point estimate and unbiased estimator (1 point): Point estimate: 4.1 inches (sample mean) It is good to note that it is an unbiased estimator of the population mean. B. Critical z-value for 95% interval (1 point): For a 95% confidence interval, z-statistic = 1.96. C. Margin of error for 95% interval (1 point): Formula: Margin of Error = z* * (σ / √n) Work: z* = 1.96 σ = 0.4 inches n = 50 Margin of Error = 1.96 * (0.4 / 7) = 1.96 * (0.4 / 7.071) = 1.96 * 0.0566 = 0.111 inches D. 95% confidence interval (2 points): Formula: x̄ ± z* * (σ / √n) Work: x̄ = 4.1 inches z* = 1.96 σ = 0.4 inches n = 50 Lower bound = = 4.1 – 1.96 × (0.4/ √ 50) = 4.1 – 0.111 = 3.989 inches Upper bound = 4.1 + 1.96 * (0.4/√50) = 4.1 + 0.111 = 4.211 inches 95% Confidence Interval: (3.989 inches, 4.211 inches) E. Sample size for 99% confidence interval with 1% margin of error (3 points): Given: 99 % confidence interval (z* = 2.576) Margin of error = 1% of point estimate = (0.01 * 4.1) = 0.041 inches - σ = 0.4 inches Formula: n = (z* * σ / E) ² Where E stands for the acceptable level of error. Work: n = (2.576 * 0.4 / 0.041) ² = (25.112 / 0.041) ² = 612.488² = 375,141.95 We are rounding up here since we require a whole number of apples to be incorporated into the bags. Minimum number of apples needed: 375,142 7. Ground Nugget Potato Company Scenario A. 95% confidence interval for mean potato weight (3 points): Given: - Sample size (n) = 100 For the same sample we have; Sample mean (x̄) = 700 grams Population standard deviation (σ) = 0.89 grams Credibly, the results are statistically significant at a 95% confidence level (z* = 1.96) Formula: x̄ ± z* * (σ / √n) Work: Margin of Error = 1.96 × (0.89 /√100) = 1.96 * (0.89 / 10) = 1.96 * 0.089 = 0.17444 grams Lower bound = 700 – 0.17444 g = 699.82556 g Upper bound = 700 + 0.17444...