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The mean, also known as the expected value, is a core concept in descriptive statistics. It gives us a central value for a set of numbers. In this particular exercise involving coffee refills at Dan's Truck Stop, the mean helps us understand the average number of refills someone might get.
To calculate the mean, we use probabilities rather than percentages. We find the probability of each event and multiply it by the number of refills it represents. Finally, we sum up these contributions. This way, we account for how each possible outcome contributes to the average.For example: - 0 refills with a probability of 0.3 is calculated as: \(0 \times 0.3 = 0\)- 1 refill with a probability of 0.4 is: \(1 \times 0.4 = 0.4\)- 2 refills with a probability of 0.2 is: \(2 \times 0.2 = 0.4\)- 3 refills with a probability of 0.1 is: \(3 \times 0.1 = 0.3\)Adding these together gives us the mean: \(0 + 0.4 + 0.4 + 0.3 = 1.1\). This means, on average, customers get 1.1 refills.

Variance is a measure of how much the values in our dataset differ from the mean. It's a crucial aspect when determining how spread out the data is. In this context, we're looking at how much deviation there is in the number of coffee refills at Dan's Truck Stop.The formula for variance takes into account each possible outcome's squared deviation from the mean, multiplying it by its probability. This quantifies the variability in terms of the squared units of the original data points.Let's break this down using our provided data:- The mean number of refills is 1.1.- Calculate the squared deviation for each count of refills: - For 0 refills: \((0 - 1.1)^2 = 1.21\) - For 1 refill: \((1 - 1.1)^2 = 0.01\) - For 2 refills: \((2 - 1.1)^2 = 0.81\) - For 3 refills: \((3 - 1.1)^2 = 3.61\)- Weight these squared deviations by their probabilities: - \(1.21 \times 0.3 = 0.363\) - \(0.01 \times 0.4 = 0.004\) - \(0.81 \times 0.2 = 0.162\) - \(3.61 \times 0.1 = 0.361\)Finally, add these to find the variance: \(0.363 + 0.004 + 0.162 + 0.361 = 0.89\). This figure represents the variability or dispersion of refill counts.

The standard deviation provides a way to express variance in the same units as the original data, making it easier to interpret. It tells us about the typical spread of the values from the mean. For Dan's Truck Stop coffee refills, the standard deviation gives insight into the usual range of refills a customer might get, up or down from the average.To find the standard deviation, simply take the square root of the variance. This reverses the squaring used in the variance calculation, providing a more intuitive understanding.Here, the variance was calculated as 0.89. Taking the square root:\(\sigma = \sqrt{0.89} \approx 0.943\)This result means that the number of coffee refills usually deviates from the mean by about 0.943 refills. It's close to 1, indicating a moderate level of variability around the average of 1.1 refills.