\( 1 \leftarrow \) A restaurant server believes the distribution of their tips has a model that is slightly skewed to the right, with a mean of \( \$ 9,60 \) and a standard deviation of \( \$ 5.40 \). Complete parts a through c below. a) Explain why you cannot determine the probability that a given party will tip them at least \( \$ 20 \). Choose the correct answer below. A. \( \$ 20 \) is \( \$ 10.40 \) greater than the mean, but \( \$ 10.40 \) is greater than the standard deviation. Since this is true and all determinable values must occur within one standard deviation of the mean, the probabilities of values that far from the mean cannot be determined. B. \( \$ 20 \) is \( \$ 10.40 \) greater than the mean, but \( \$ 10,40 \) less than the mean is \( -\$ 0.80 \). Since these values should have the same probability and tipping a negative amount is not possible, the probabilities of values that far from the mean cannot be determined. C. This distribution is not unimodal, meaning it has more than one mode and does not meet the conditions of the Normal model, so the probabilities of values within this distribution cannot be determined. D. This distribution is skewed, meaning it is non-symmetric and does not meet the conditions for the Normal model, so the probabilities of values within this distribution cannot be determined.
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The correct answer is D. Since the distribution of tips is slightly skewed to the right, it indicates that the data points are not symmetrically distributed around the mean. In a normal distribution, probabilities can be easily determined, but because this distribution is non-symmetric, we cannot accurately utilize the standard normal probabilities. This means we can't directly determine the probability of a party tipping at least $20. To clarify, in a skewed distribution, a larger tail on one side can significantly affect probabilities, making it tricky to apply standard normal distribution tables or methods. Thus, any calculations relying on those concepts may yield misleading results, particularly for values that are more extreme, such as the $20 tip in this case.
