QuestionLet denote the number of broken eggs in a randomly selected carton of one dozen eggs. \begin{tabular}{|c|c|c|c|c|c|} \hline & 0 & 1 & 2 & 3 & 4 \\ \hline & 0.60 & 0.25 & 0.10 & 0.03 & 0.02 \\ \hline \end{tabular} (a) Calculate and interpret . (b) Consider the following questions. (i) In the long run, for what percentage of cartons is the number of broken eggs less than ? \% (ii) Does this surprise you? Yes No (c) Explain why is not equal to . This computation of the mean is incorrect because it does not take into account that the number of broken eggs are all equally likely. This computation of the mean is incorrect because the value in the denominator should equal the maximum value. This computation of the mean is incorrect because it does not take into account the number of partially broken eggs. This computation of the mean is incorrect because it includes zero in the numerator which should not be taken into account when calculating the mean. This computation of the mean is incorrect because it does not take into account the probabilities with which the number of broken eggs need to be weighted.
Studdy Solution
STEP 1
1. The random variable represents the number of broken eggs in a carton.
2. The probability distribution of is given in the table.
3. The expected value is calculated as the weighted average of the possible values of .
STEP 2
1. Calculate the expected value .
2. Interpret the value of .
3. Determine the percentage of cartons with fewer broken eggs than .
4. Explain why is not equal to the arithmetic mean of the values.
STEP 3
Calculate the expected value using the formula:
Substitute the values from the table:
Calculate each term:
Add the terms:
STEP 4
Interpret the value of :
The expected number of broken eggs in a randomly selected carton is .
STEP 5
Determine the percentage of cartons with fewer broken eggs than :
Identify the values of that are less than : .
Calculate the probability:
Convert to percentage:
STEP 6
Explain why is not equal to :
This computation of the mean is incorrect because it does not take into account the probabilities with which the number of broken eggs need to be weighted.
The expected value is:
Percentage of cartons with fewer broken eggs than :
Explanation for why the initial mean calculation is incorrect:
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