For this question, we have to develop a model in Excel which lists all the constraints, decision variables, and objective variable. Following is the developed model:

Here, the decision variables are highlighted in yellow, constraints in red, and objective function in green.

Decision variables are the units of each type of widget, constraints are imposed on the labour hours per week, and objective function is set to the Total profit made each week. Units of widget are given some seed values, such as 1 and 2.

Our main objective is to maximize total profit by moving the decision variables subject to the constraints.

**Part 1A**

We have to determine how many units of widget A and widget B must be produced each week to maximize the total profit.

The model can be described as below:

Maximize 70x + 50y, subject to the following constraints:

2x + 4y <= 100

5x + 3y <= 110

Where, x is the units of Widget A and y is the units of Widget B.

For this, we can use the Solver by setting parameters as shown below:

Then, use the Solve button to find the required number of units for the widgets and the total profit.

After Solving, we get the following results:

- Units of widget A = 10
- Units of widget B = 20
- Maximum Profit = $1,700.00

**Part 1B**

We have to determine the new values when at least 16 units of widget A is produced per week.

The new model can be described as below:

Maximize 70x + 50y, subject to the following constraints:

2x + 4y <= 100

5x + 3y <= 110

x >= 16

Where, x is the units of Widget A and y is the units of Widget B.

For this, we can use the Solver by setting parameters as shown below:

Then, use the Solve button to find the required number of units for the widgets and the total profit.

After Solving, we get the following results:

We have to determine the new values when the company estimates that at most 25 units of widgets can be handled by its distribution network each week along with the above constraints.

The new model can be described as below:

Maximize 70x + 50y, subject to the following constraints:

2x + 4y <= 100

5x + 3y <= 110

x >= 16

x + y <= 25

Where, x is the units of Widget A and y is the units of Widget B.

For this, we can use the Solver by setting parameters as shown below:

Then, use the Solve button to find the required number of units for the widgets and the total profit.

After Solving, we get the following results:

- Units of widget A = 17.5
- Units of widget B = 7.5
- Maximum Profit = $1,600.00

**Question 2**

**Part 2A**

- Consider the following line plot generated for the temperature anomalies over the period of 1880 to 2016:

As we can see from the plot, there is clearly a positive trend observed in the plotted data, where there is a steady increase in the temperature anomalies over the 1880-2016 time period.

This can be treated as a sign of global warming, since the temperature steadily increased over the years. - A one-sample t-test was conducted on the average temperature anomalies data. Finally, we can interpret the results at 5% significance level.

Following is the result of the one-sample t-test:

The dataset was subjected to the one-sample t-test, and the outcome shows a strong evidence opposing the null hypothesis. With a relatively low p-value of 0.0003874, the sample mean (0.04462022) is substantially larger than zero. As a result, we can conclude that there’s statistical support for the alternative hypothesis that the actual population’s mean is larger than 0, which would indicate an increase in temperature anomalies or an upward trend in temperature. - The scatter plots and the results for each of the 4 time periods is shown below.

GlobalA2 data

This period’s R-squared value is 0.1793, indicating a slight linear connection between time and temperature. So, we can say that there was a slight global warming in this period.

Global A3 data

There is hardly any proof that the temperature anomalies over this time have been trending significantly, as seen by an almost straight horizontal trend line in the plot.

Thus, there is no conclusive proof that global warming occurred during this time.

Global A4 data

Temperature anomalies within this time period show an upward trend and it is more stronger compared to other time periods. This evidence points to global warming at this time period.

Global A5

The Global A5 [1998-2016] era has some evidence of global warming, although the intensity of the evidence is less than it is for some other periods, as seen by the comparatively low R-squared value.

**Part 2B**

- The plot showing relationship between temperature anomalies and CO2 levels is shown below:

Intercept of the model is -5.0641 which does not have meaningful inference because it is nearly impossible for CO2 levels to become zero. Slope coefficient of the estimated trend line is 0.0156, which means that the temperature anomaly increases approximately 0.0156 °C for a one-unit increase in CO2 level (ppm). The R-square value of the model is 0.6123, which is relatively high. This means that about 61.23% of the variation is explained by changes in CO2 level, which indicates a reasonably good fit for the simple linear regression model.

The association between CO2 concentrations and variations in temperature is statistically significant, as shown by the low p-value ( 2.2e-16) associated with the CO2 coefficient.

We can use the predict function to predict the temperature anomaly for CO2 = 400 ppm.

The result is obtained as:

Thus, Predicted Temperature Anomaly for CO2 level = 400ppm is 1.166 °C

**Part 2C**

**Introduction**

Both the scientific community and the general public have been deeply interested in the issue regarding if Earth is warming and if CO2 emissions are to blame. By examining temperature anomalies and CO2 levels throughout a range of historical periods, we hope to provide answers to these important concerns in this research. We will look for patterns and connections in the data using data analytics approaches.

**Data Analytics Techniques Used**

To answer these problems, we applied a number of data analytics approaches to anomalies in temperature and CO2 level data:

- We started with the analysis of temperature anomalies during the period of 1880 to 2016 using line plots and one-sample t-test. With the use of this technique, we were able to determine if there is proof of global warming and how important temperature changes are. The main method utilised was a one-sample t-test for hypothesis testing.
- We looked more closely at temperature anomalies for four different time periods. To determine the intensity and relevance of the trends, we evaluated R-squared values and applied linear regression.
- We used straightforward linear regression to examine the connection between CO2 levels and temperature anomalies. As a result, we were able to forecast temperature anomalies for particular CO2 levels and evaluate the effect of CO2 on temperature.

**Results and Discussion**

From 1880 to 2016, we saw a definite upward trend in temperature anomalies, which indicates global warming. The one-sample t-test demonstrated that temperature anomalies increased statistically significantly with time (p-value = 0.0003874). In Part 2B, different eras of time yielded differing levels of proof of global warming. Global A2 demonstrated a lesser positive trend, Global A4 and Global A5 had moderate positive trends. There was no discernible trend in Global A3.

The CO2 study (Part 2B) showed a correlation between temperature anomalies and CO2 levels that was favourable.

The model’s R-squared value of 0.6123 showed that variations in CO2 levels could account for around 61.23% of the variance in temperature anomalies. The CO2 coefficient’s low p-value ( 2.2e-16), which indicates statistical significance, supported this association.

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