Questions: 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.

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.
Transcript text: 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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Solution

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Solution Steps

Step 1: Explanation of the Distribution

The server believes that the distribution of tips is slightly skewed to the right, which indicates that it is non-symmetric. Therefore, it does not meet the conditions for the Normal model. As a result, we cannot determine the probability that a given party will tip at least \$20. The correct answer is:

\[ \text{D. This distribution is skewed, meaning it is non-symmetric and does not meet the conditions for the Normal model, so the probabilities of values within this distribution cannot be determined.} \]

Step 2: Calculation of Z-scores

To find the probability that a given party will tip at least \$20, we first calculate the Z-score for \$20 using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

where:

  • \(X = 20\)
  • \(\mu = 9.60\)
  • \(\sigma = 5.40\)

Calculating the Z-score:

\[ Z_{start} = \frac{20 - 9.60}{5.40} \approx 1.9259 \]

The Z-score for the upper bound (infinity) is:

\[ Z_{end} = \infty \]

Step 3: Probability Calculation

Using the Z-scores, we can express the probability that a given party will tip at least \$20 as:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(1.9259) \]

From the calculations, we find:

\[ P \approx 0.0271 \]

Final Answer

The probability that a given party will tip at least \$20 is approximately \(0.0271\).

Thus, the final answer is:

\[ \boxed{P \approx 0.0271} \]

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