Baseball There were 116 World Series from 1903 to 2020. Use th...

Question

Baseball There were 116 World Series from 1903 to 2020. Use the probability distribution in Exercise 30 to find the number of World Series that had 4, 5, 6, 7, and 8 games. Find the population mean, variance, and standard deviation of the data using the traditional definitions. Compare to your answers in Exercise 30.

Accepted Answer

Welcome back, everyone. A quiz competition can last 123, or 4 rounds. Over 50 competitions. The probability distribution for the number of rounds is as follows PF1 equals 0.180, P 2 equals 0.320, PF 3 equals 0.340, and P 4 equals 0.160. Based on this distribution, calculate the population mean variance and standard deviation for the number of rounds played. So first of all, we're going to begin with our population mean, which is simply the expected value E of X. Let's recall the formula. We basically have to sum. Probabilities multiplied by the corresponding random variable values. So, first of all, we take the first value of 1, which is multiplied by its corresponding probability of 0.180. We then add 2 multiplied by its probability of 0.32 0. Followed by 3, multiplied by 0.340. And 4 multiplied by 0.160. Performing the calculation, we end up with 2.5. That would be our mean. And then we are ready to proceed with our other values. So that'd be variance, sigma squared, and we can recall the formula. We take the second row of moment E of X2. And we subtract the expected value squared, so e of X. Squared So now let's identify the second row of moments, that would be E of X squared. And what we have to do is simply square our random variable X values and multiply them by the corresponding probabilities, right? So we're going to sum. Probabilities multiplied by the corresponding x values squared. So that would be 1 squared multiplied by 0.180 plus 2 squared multiplied by 0.320 plus. 3 squared multiplied by. 0.340 and plus 4 squared multiplied by 0.160. Which is equal to 7.1. And how to identify the variants, we're going to use our formula, right? So, sigma squared is going to be equal to. The second raw moment there'll be 7.1. We can essentially not round it to two decimal places, so that would be 7.08. And we're subtracting the expected value, so we can also round it to two decimal places, that would be 2.48, right? So we're subtracting 2.48 squared. And that will be equal to 0.930. And now from here, all that we have to do is identify the standard deviation sigma. Which is square root of variance. And that'll be square root off. 0.930 Which is equal to 0. 9 65. Well, then we have our answers, let's stay them. So the mean value is equal to 2.48. Our variance sigma squared is 0.930, and our standard deviation is 0.965. Those will be our final answers and thank you for watching.

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Key Concepts

Probability Distribution

Population Mean

Variance and Standard Deviation

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