Interpret The Results In Part I And Part II - 25 Points

Interpret The Results In Part I And Part Iipart I 25 Pointsfind Th

Interpret the results in Part I and Part II. Part I (25 points) involves calculating specific areas under the standard normal curve and applying normal distribution concepts to real-life scenarios. Part II (15 points) includes constructing confidence intervals for sample means and determining the minimum sample size required for estimation. Part III (30 points) requires conducting a hypothesis test about the caffeine content in cola bottles. Part IV (30 points) involves performing regression analysis on insurance data to analyze the relationship between the number of employees and administrative costs.

Paper For Above instruction

Introduction

Statistical analysis plays a vital role in making informed decisions across various fields, including manufacturing, finance, healthcare, and insurance. This paper interprets the results from four parts of a statistical problem set, emphasizing normal distribution calculations, confidence intervals, hypothesis testing, and regression analysis. Each section demonstrates different statistical concepts, critical for understanding data and deriving meaningful conclusions.

Part I: Normal Distribution Calculations

In Part I, the focus is on applying properties of the standard normal distribution. The problem asks to find the area under the normal curve to specific regions: to the right of z = 0.65, to the left of z = -2.13, and between z = -0.34 and z = 0.62. These values can be obtained using standard normal distribution tables or statistical software. For instance, the area to the right of z = 0.65 corresponds to 1 - P(Z

For the interval between z = -0.34 and z = 0.62, we calculate P(Z

The tire store scenario applies the normal distribution to a practical problem. Given the mean tire lifespan of 26,640 miles and a standard deviation of 4,000 miles, the question is how many of the 9,000 tires will last between 25,000 and 30,000 miles. Standardizing these bounds involves computing z-scores: for 25,000 miles, z = (25,000 - 26,640)/4000 ≈ -0.41; for 30,000 miles, z ≈ 0.35. Using the standard normal table, P(Z

Part II: Confidence Intervals and Sample Size Determination

Part II involves constructing confidence intervals for the mean closing price of Apple stock based on a sample of 36 days. The sample mean is $116.16 with a standard deviation of $10.27. To estimate the population mean with a specified level of confidence, the margin of error (ME) is calculated using the z-value corresponding to the confidence level and the standard error.

For a 90% confidence interval, the critical z-value is approximately 1.645. The standard error (SE) is 10.27 / √36 = 10.27 / 6 ≈ 1.711. The margin of error is ME = 1.645 * 1.711 ≈ 2.813. Therefore, the 90% confidence interval is ($116.16 - 2.813, $116.16 + 2.813) ≈ ($113.35, $118.97).

For a 95% confidence interval, the critical z-value is approximately 1.96. Using the same SE, ME = 1.96 * 1.711 ≈ 3.356. The interval is ($116.16 - 3.356, $116.16 + 3.356) ≈ ($112.80, $119.52).

Between the two, the 95% confidence interval is wider, reflecting a higher degree of certainty that the interval contains the true mean.

Next, to determine the minimum sample size (n) required to estimate the population mean within one unit (d=1) with 95% confidence, given a population standard deviation of σ=4.8, we utilize the formula: n = (Z σ / d)^2. For 95%, Z ≈ 1.96. Plugging in the values, n = (1.96 4.8 / 1)^2 ≈ (9.408)^2 ≈ 88.54. Since sample size must be an integer, at least 89 observations are necessary to ensure that the estimate is within one unit of the true mean with 95% confidence.

Part III: Hypothesis Testing on Caffeine Content

The hypothesis test aims to verify the company's claim that the mean caffeine content per 12-ounce bottle is 40 mg. The null hypothesis (H₀) is that the mean is 40 mg, while the alternative hypothesis (H₁) considers the possibility that the mean differs from 40 mg.

Given a sample of 20 bottles with a mean caffeine content of 39.2 mg, and knowing the population standard deviation (σ) is 7.5 mg, the test uses a significance level α=0.01.

Calculating the standardized test statistic:

z = (x̄ - μ₀) / (σ / √n) = (39.2 - 40) / (7.5 / √20) ≈ -0.8 / (7.5 / 4.4721) ≈ -0.8 / 1.677 ≈ -0.477.

The critical value for a two-tailed test at α=0.01 is approximately ±2.576. Since the calculated z of -0.477 falls well within the acceptance region (-2.576, 2.576), we fail to reject H₀.

The p-value, representing the probability of observing such a result or more extreme under H₀, is approximately 0.634 (two-sided). The high p-value indicates insufficient evidence to reject the company's claim that the mean caffeine content is 40 mg.

In conclusion, the test does not provide sufficient statistical evidence at the 1% significance level to refute the company's statement.

Part IV: Regression Analysis Between Employees and Administrative Costs

Part IV involves analyzing the relationship between the number of employees (x) and administrative costs (y). Using the provided data, calculations of sums: Σx, Σy, Σx², Σy², and Σxy are performed to determine the strength and nature of the correlation.

With these calculations, the correlation coefficient (r) is obtained, measuring the strength and direction of the linear relationship. A positive r indicates that as the number of employees increases, administrative costs tend to increase. The regression line is calculated with the least squares method, providing an equation of the form y = a + bx, where b is the slope, and a is the intercept.

The computed slope indicates the expected change in y for a one-unit increase in x. A positive slope confirms a positive relationship, suggesting that larger employee groups tend to incur higher administrative costs. The correlation's magnitude indicates whether the relationship is weak, moderate, or strong—typically, |r| > 0.7 signifies a strong correlation.

For prediction purposes, plugging x = 10 into the regression equation yields an estimated y, the average administrative cost corresponding to a group of ten employees. Such predictions assist in budgeting and resource planning within organizational contexts.

Conclusion

In summary, these analyses demonstrate fundamental statistical tools: calculating probabilities under the normal distribution, constructing confidence intervals, conducting hypothesis testing, and implementing regression analysis. Each method provides critical insights into data behaviors, helping organizations and researchers make data-driven decisions with quantifiable confidence.

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