Finding the Mean, Variance, and Standard Deviation In Exercise...

Question

Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

Dogs The number of dogs per household in a neighborhood

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Accepted Answer

Welcome back, everyone. In this problem, the number of smartphones owned per student in a college dormitory is recorded as follows. Calculate the mean variance and the standard deviation of this probability distribution. Now, in order to help us figure out each of these measures, it would be a good idea to use a table to help us organize the calculation because we'll need to find a few sums from our data. For example, we're supposed to calculate the mean, right, but recall, OK, that the mean. For this probability distribution or for a probability distribution mu. Is equal to the sum of values multiplied by their probabilities. So in that case, it would be a good idea for us to create a table with the values and their probabilities. So let's go ahead and do that. So let's say here first we have our value X. Now make this table a little longer because as I said, we'll be coming back to it eventually for the variance and the standard distribution. And here we have our probabilities for each value. So now, let's find a product of the value and its probability. X multiplied by P of X. So for example, when X is 0, it's probability is 0.018. When it's 1, it's 0.842. Then 2 and 0.096, 3 and 0.032, and 4, OK, and 0.012. So now we need to find each of these products. So for example, for the first one. 0 multiplied by 0.018 is 01 multiplied by 0.842 is 0.842. 2 multiplied by 0.096 is 0.192. 3 multiplied by 0.032 is 0.096, and 4 multiplied by 0.012 is 0.048. So now we need to find a sum here, OK, of these values. So when we add all of our products. OK, or add all for. Yes, add all of our products. Let me put that in red here, OK. Then for our sum, we should find that when we add all of these up, it's equal to 1.178. OK? So the mean is going to be equal to 1.178 or approximately 1.2. Next, we need to find a variant. What is variance? Well again recall that the variance. And for probability distribution, sigma squared is equal to the sum of the squared deviation, so that the sum of X minus mu or squared multiplied by their probabilities p of X. So again, to help us find these variants, then let's see if we can extend this for this sum, OK? So we already know what the mean is, but now, let's add on to our table a column that represents our square or sorry, our deviations X minus mu. That is how far each value is away from the mean. Then let's square those deviations X minus mu squared, OK. And then let's multiply those results by the, the, the probabilities for each deviation. So that's X minus mu squared multiplied by PX probability, OK? The probability of X. Now, let's take the first one. So we already know that our mean value is 1.178, OK? So, when it comes to the first one for the first value is 0, our deviation is going to be 0 minus 1.178, which is gonna be -1.178. When we squared that, it's equal to 1.3877, and when we multiply that square by our probability, We get 0.024978. Essentially, this is what we're going to do for the rest of values. So when X equals 1, then our difference is negative 0.178. The squared deviation is 0.0317, and the product is 0.026678. When X equals 2, then we're gonna get 0.822, 0.6757, and 0.064866 respectively. When it's 3, we get 1.822, 3.3197, and 0.106230. And when it's 4, we get 2.822, 7.9637, and 0.095564. So now we have all the products of the squared deviations and their probabilities. So again. If we were to add all of these up from our table, OK. Then we should find that we get this sum, OK, to be equal to. To be equal to 0.318316, OK? So that means our variance is going to be equal to 0.318316 or approximately 0.3. Know that we have the variants, we can try to find our standard deviation. And I recall that the standard deviation sigma. Is really the square root of the variant. We already know our value for the variance, so then that's gonna be the square root of 0.318, 316, and that is approximately 0.6, OK? So based on our data, what can we conclude? Well, now this tells us that our mean mu is approximately 1.2. Our our variance is approximately 0.3. And our standard deviation is approximately 0.6. Thanks a lot for watching everyone. I hope this video helped.

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Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.
Dogs The number of dogs per household in a neighborhood

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Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.Dogs The number of dogs per household in a neighborhood

Watch next

��app

Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.
Dogs The number of dogs per household in a neighborhood