In Exercises 5–20, find the range, variance, and standard devi...

Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Smart Thermostats Listed below are selling prices (dollars) of smart thermostats tested by Consumer Reports magazine. Are any of the resulting statistics helpful in selecting a smart thermostat for purchase?

250 170 225 100 250 250 130 200 150 250 170 200 180 250

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says listed below are the battery life durations and hours of different smartphone models tested by a consumer research agency. Find the range variants of standard deviation of the given sample data. Include appropriate units such as hours in your results, then answer the given question. Are any of the resulting statistics helpful in selecting a smartphone based on battery life? And for our data, we're given the battery lives in hours of 1012, 1416, and 18. We're also given four possible choices as our answers. For choice A, we have the range is equal to 10 hours, the variance is equal to 14 hours squared. The standard deviation is equal to 4.31 hours, and yes, the range alone is enough to determine the best smartphone model. For choice B, we have the range is equal to 8 hours, the variance is equal to 10 squared, the standard deviation is equal to 4.31 hours, and yes, the standard deviation helps identify which smartphone has the longest battery life. For choice C, we have the range is equal to 8 hours, the variance is equal to 10 hours squared. The standard deviation is equal to 3.16 hours. And no, the calculated statistics only describe the variation in battery life, but do not directly help in selecting a specific smartphone. And for choice D, we have the range is equal to 10 hours, the variance is equal to 14 hours squared. The standard deviation is equal to 3.16 hours, and no, because statistical measures like variance and standard deviation are not useful for comparing real-world smartphones' performance. So, the first thing we need to do is actually calculate the range, the variance, and standard deviation of our data. So, we're gonna start off with the range. And recall that the range is equal to the maximum value. Minus the minimum value. Which in this case, if I look at my data points, the maximum value is gonna be 18 hours, and I'll subtract the minimum value, which is 10 hours, which means my range is going to be equal to 8 hours. So, that is the first quantity that we needed to calculate. Now for the second quantity, we'll need to calculate our variance, which is Squad, and here we call your formula for the variance, and that's going to be equal to the sum of the quantity of X of I minus X in quantity squared divided by the quantity of N minus 1. Here X of I is one of our individual data points. X bar is our mean, and N is the number of data points that we have, which in this case is equal to 5. So, in order to use a formula, we'll first need to calculate our mean. So, we need to calculate X bar, and recall your formula for calculating the mean of a data set, that's just going to be adding up all of our data points, and then divided by the number of data points that we have. So, X bar is going to be equal to. 10 + 12 + 14 + 16 + 18, and that whole quantity is divided by the number of data points that we have, which is 5. And when we evaluate this expression, we see that X bar is going to be equal to. 14 hours. So now we have enough information to actually use our formula to calculate our variance. So, S2 is going to be equal to. In the numerator, I'll have the quantity for the first term, I'll have 10 minus 14 in quantity squared. And plus The quantity of 12 minus 14 in quantity squared. Plus, the quantity of 14 minus 14 in quantity squared. Plus, the quantity of 16 minus 14 in quantity squared. Plus the quantity of 18 minus 14 in quantity squared, and all of that is divided by the quantity of N minus 1, which in this case is going to be 5 minus 1. So simplifying this expression, this is going to be equal to. For the first term of the numerator, I'll have 10 minus 14, which will give you minus 4, and we square that, we get 16. And we'll have plus For the next term, I'll have 12 minus 14, which will give you -2, and squaring that will give you 4. Then we'll have plus, for the next term, we'll have 14 minus 14, which is 0, and squaring 0 will give me 0 as well. Then we'll have plus. Or the next term will have 16 minus 14, which is 2, and when I square 2, I get 4. And then we'll have less for the last term, we'll have 18 minus 14, which is 4, squaring 4 will give me 16. And all of that is divided by the quantity of 5 minus 1, which is 4. So evaluating this this expression, we will get. 10 and that has units of hours squared. Since in the numerator we have the quantity of X of minus X bar, both our individual data points, as well as our mean, have units of hours, and when we square that, we get units of hours squared. So, this is the 2nd quantity that we needed to calculate. And for the third quantity, and our last quantity that we need to calculate, that is the standard deviation, which is S, and that's equal to the square root of our variance Squad. So this is going to be equal to the square root of 10, which is approximately equal to. 3.16, and this has units of hours. That's because our variance has units of our squared, and we're taking the square root of that numerical value, we also need to take the square root of units. They're taking the square root of our squared gives me units of hours. So this here is the final quantity that we needed to calculate. So now we just need to answer our final question, which are any of the resulting statistics helpful in selecting a smartphone based on battery life? Now, our calculated statistics, which are the range variance and standard deviation, provide insight into how much battery life varies among the smartphone models. However, they may not be directly useful for selecting a specific smartphone. So, if we look for reasons why this might be the case, first off is the range. It shows the difference between the longest and shortest battery life, but it doesn't indicate which models have better performance. The variance and standard deviation describe the spread of battery life values, but do not identify the best or worst performing model. So while these statistics help understand the variation in battery life across different models, they do not directly tell which specific smartphone is the best choice. So, taking that into account, as well as the values that we calculated for the range, the variance and standard deviation, if we look at our answer choices. We see that the correct answer choice is answer choice C. Since the values for the range, the variance and standard deviation match what we calculated below. And it has the correct answer for our question. So, I hope this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Range

Variance

Standard Deviation

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