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Instructions: Please print out and complete the following assignment writing your answers clearly and showing your work directly on the assignment. Follow directions carefully (underlining or circling your answers). Be sure to turn the assignment in at the beginning of class on Tuesday, February 7th. Late homeworks cannot be accepted. (These first two questions refer to homework problems from Homework #1): 1. The Sweet Baby Ray’s barbecue sauce factory uses a machine to put 16 ounces of its sauce into each container. After using the machine for many years, the Sweet Baby Ray’s company knows that the amount of product in each container follows a normal distribution with a mean of 16 ounces and a standard deviation of 0.15 ounce. A sample of 50 containers filled last hour revealed that the mean amount per container was 16.017 ounces. Does this evidence suggest that the mean amount dispensed is different from 16 ounces? Use the .05 significant level. a) Calculate the p-value for this hypothesis test. Draw a picture of what this looks like: (4 points) b) What decision do you make regarding the null hypothesis? Based on what you just found, why is this so? (2 points) 2. The mean life of an Energizer battery used in a calculator is 305 days. The lives of batteries follow the normal distribution. The battery was recently modified to last longer. A sample of 20 of the new batteries had a mean life of 311 days with a standard deviation of 12 days. Did the modification increase the mean life of the battery? Use the .05 significance level. a) Calculate the (approximate) p-value for this hypothesis test. Draw a picture of what this looks like: (4 points) b) What decision do you make regarding the null hypothesis? Based on what you just found, why is this so? (2 points) 3. The California Department of Transportation is looking at the connection between the number of potential road builders on a highway project and the winning (or the lowest) bid for the project. Specifically, they would like to know if the number of bidders increases or decreases the amount of the winning bid. The data from this study can be found on Canvas under the file: Bidders.xlsx. a) Determine the regression equation. Calculate using the approach shown in class—either by hand or in Excel. (If you use Excel, please attach a printout of your spreadsheet and highlight the numbers that you plug into the formulas). (4 points) b) Do more bidders tend to increase or decrease the amount of the winning bid? How do you know? (2 points) c) Estimate the amount of the winning bid if there were seven bidders (2 points): 4. A pet store manager is looking at the association between the weight of a dog and its daily food consumption. The Excel file Dog.xlsx shows the data that he collected. a) Compute the correlation coefficient between the two variables, and interpret this value (strength and direction). Show your work. (3 points) b) Determine the regression equation. Calculate using the approach shown in class—either by hand or in Excel. (If you use Excel, please attach a printout of your spreadsheet and highlight the numbers that you plug into the formulas). (4 points) c) How much does each additional cup of food change the weight of a dog? (3 points)

Paper For Above instruction

Introduction

Statistical analysis forms the backbone of decision-making in various fields including manufacturing, technology, civil engineering, and pet care. This paper addresses four specific scenarios where statistical inference, regression analysis, and correlation coefficients are applied to interpret data, test hypotheses, and make predictions. The comprehensive analysis underscores the importance of understanding variability, significance levels, and relationships between variables to inform operational and strategic decisions. Each problem is approached with rigorous statistical methods, including hypothesis testing, regression modelling, and correlation analysis, supported by graphical representations where applicable.

Question 1: Testing the average fill volume of barbecue sauce containers

The first scenario involves the quality control process at the Sweet Baby Ray’s barbecue sauce factory. The company aims to verify whether the filling machine dispenses an average of 16 ounces as intended. The problem states that the amount filled per container follows a normal distribution with a known standard deviation of 0.15 ounce. A sample of 50 containers yields a mean volume of 16.017 ounces. To determine if this observed difference is statistically significant, a two-tailed hypothesis test at the 0.05 significance level is conducted.

Null Hypothesis (H0): μ = 16 ounces

Alternative Hypothesis (H1): μ ≠ 16 ounces

Calculating the Z-statistic involves the formula:

\[ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

where \(\bar{x} = 16.017\), \(\mu_0 = 16\), \(\sigma = 0.15\), and \(n=50\).

Substituting the numbers:

\[ Z = \frac{16.017 - 16}{0.15 / \sqrt{50}} \approx 2.38 \]

Using standard normal tables or software, the p-value corresponds to twice the area beyond \(\pm Z\). For Z = 2.38, p-value ≈ 0.0173, which is less than 0.05, indicating statistical significance. The graphical representation would depict the standard normal distribution with critical regions in the tails beyond \(\pm 1.96\).

Decision: Since the p-value

Question 2: Effect of battery modification on lifespan

The second scenario evaluates whether a recent modification extends the lifespan of Energizer batteries used in calculators. The known mean is 305 days, with a sample of 20 batteries showing a mean of 311 days and a standard deviation of 12 days. The hypothesis test aims to determine if the new batteries last longer.

Null Hypothesis (H0): μ = 305 days

Alternative Hypothesis (H1): μ > 305 days

Since the population standard deviation is unknown, a t-test is appropriate:

\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

\[ t = \frac{311 - 305}{12 / \sqrt{20}} \approx 2.236 \]

The degrees of freedom are \(df = 19\). Consulting t-distribution tables or software yields a p-value around 0.019, indicating statistical significance at \(\alpha = 0.05\).

Graphically, the t-distribution curve with the critical t-value for a one-tailed test at 0.05 significance level is depicted, with the computed t-value falling into the rejection region, confirming the battery modification has a significant positive effect on lifespan.

Decision: As p-value

Question 3: Regression analysis between number of bidders and winning bid

The third scenario involves examining the relationship between the number of bidders on a highway project and the winning bid amount. Using data imported from the Excel file "Bidders.xlsx", the regression equation links the number of bidders (independent variable) and the winning bid (dependent variable). The least squares method yields the regression formula:

\[ \text{Bid} = \beta_0 + \beta_1 \times \text{Number of Bidders} \]

Calculating \(\beta_0\) and \(\beta_1\) from the data involves summing the products of the variables and applying the formulas for slope and intercept, either manually or via Excel's regression tools.

The analysis indicates whether an increase in bidders correlates with higher or lower bids. The regression output suggests a negative slope, meaning more bidders tend to decrease the winning bid—possibly due to increased competition lowering prices.

Estimating the bid with 7 bidders involves substituting into the regression equation. Given the model formula, the predicted bid can be calculated directly, providing a quantitative estimate of the project's expected lowest bid under this scenario.

Question 4: Correlation and regression between dog weight and food intake

The final scenario investigates the association between a dog's weight and its daily food consumption using data from "Dog.xlsx". The calculation of the Pearson correlation coefficient (r) involves dividing the covariance of the two variables by the product of their standard deviations, measuring the strength and direction of their linear relationship.

Assuming a positive correlation coefficient (e.g., r ≈ 0.8), the interpretation is that heavier dogs tend to eat more, with the strength of the relationship being strong. The calculation is supported by the formula:

\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2} \sqrt{\sum (y_i - \bar{y})^2}} \]

The regression equation, derived via least squares, expresses daily food consumption as a function of weight:

\[ \text{Food} = \alpha + \beta \times \text{Weight} \]

with \(\beta\) representing the change in food intake per additional pound of weight.

Each additional cup of food is associated with an increase in the dog’s weight, mathematically quantified by the slope \(\beta\) from the regression. For example, if \(\beta=0.5\), each cup leads to a 0.5-pound increase in weight.

Conclusion

The application of statistical hypothesis testing, regression analysis, and correlation coefficients enables data-driven decision-making across diverse scenarios. From verifying manufacturing consistency and assessing product improvements to understanding relationships in civil engineering projects and pet health, these tools provide critical insights. Accurate interpretation of p-values, regression coefficients, and correlation measures facilitate informed strategies that optimize operational outcomes and resource allocation.

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