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Chapter 17: Problem 3

Tips A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of \(\$ 9.60\) and a standard deviation of \(\$ 5.40 .\) a. Explain why you cannot determine the probability that a given party will tip him at least \(\$ 20\). b. Can you estimate the probability that the next 4 parties will tip an average of at least \(\$ 15 ?\) Explain. c. Is it likely that his 10 parties today will tip an average of at least \(\$ 15 ?\) Explain.

Short Answer

Expert verified
a. We can't calculate the probability that a single party will tip at least $20 as the specific probability distribution is unknown. b. There's a 2.28% chance that the next four parties will tip an average of at least $15. c. There's a 0.08% chance that ten parties today will tip an average of at least $15, which is extremely unlikely.

Step by step solution

01

Understand Probability Distribution

For a single party, we know mean and standard deviation of tip distribution, but we don't know the probability distribution itself. As the distribution is stated to be slightly skewed to the right, it doesn't follow a normal distribution. Without knowing the specific distribution or the shape of the distribution, we can't calculate the exact probability that a given party will tip at least $20.
02

Apply Central Limit Theorem for Next Four Parties

Central Limit Theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution. Here, we will consider 4 parties as one group. The mean (渭) and standard deviation (蟽) of the individual parties remain the same. But when we consider groups of parties, the standard deviation of the mean of these groups (蟽_group) would be the standard deviation divided by the square root of the number of parties in the group. Hence, for four parties, 蟽_group = \$5.4 / sqrt(4) = \$2.7. Normalizing the tip amount of 15 dollars, we'll get Z = (15 - 9.6) / 2.7 = 2. This would correspond to the 97.72 percentile of the normal distribution. So, there is a about 2.28% chance (which is 100 - 97.72) that the average tip from 4 parties will be at least $15. Hence, it's unlikely, but possible.
03

Apply Central Limit Theorem for Ten Parties Today

Again, we can apply the Central Limit Theorem. For ten parties, 蟽_group = \$5.4 / sqrt(10) = \$1.71. Normalizing the tip amount of 15 dollars, we'll get Z = (15 - 9.6) / 1.71 = 3.16. This value is even larger than before, corresponding to the 99.92 percentile of the normal distribution. There is about a 0.08% chance (which is 100 -99.92) that the average tip from 10 parties will be at least $15. Hence, it's extremely unlikely.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution
Probability distribution describes how the values of a random variable are distributed. It essentially provides the probabilities of different possible outcomes of an event. In the context of the waiter's tips, knowing the specific probability distribution would allow us to calculate the likelihood of any given outcome, such as earning at least $20 from a party. However, if the probability distribution is not fully defined, as in the waiter's case where we only know it is slightly skewed to the right, it's missing critical information that prohibits precise probability calculations. This is because the distribution's shape impacts how probabilities are assigned to potential outcomes.
Standard Deviation
Standard deviation is a measure of how spread out numbers are in a dataset. It reflects how much variation there is from the average (mean). For the tips example, the standard deviation of $5.40 indicates that individual tip amounts generally differ by this amount from the average tip of $9.60. This measure of spread is crucial when assessing variability. A large standard deviation suggests a lot of variability in tips, whereas a small standard deviation indicates the tips are close to the mean. This variability measure becomes especially relevant when applying the Central Limit Theorem to larger groups, as it helps to normalize and interpret data by adjusting for population size.
Normal Distribution
A normal distribution is a probability distribution characterized by a symmetric, bell-shaped curve. It is defined by two parameters: the mean and the standard deviation. While the waiter's tips are not perfectly normally distributed due to the right skew, the Central Limit Theorem allows the approximation of a normal distribution for the average of a large number of tips. This is why the waiter's tips for groups of parties can still utilize normal distribution models, despite the skew. Using the mean and adjusted standard deviation for groups of parties, we model and predict probabilities, like the chance of tips averaging above a certain amount.
Mean
The mean, also known as the average, is the central value of a set of numbers. It is calculated by adding all numbers in a dataset and dividing by the count of numbers. In this waiter scenario, the mean tip amount is $9.60. The mean is used as a central measure to derive other statistical values and to predict outcomes. In the context of groups of tips, the mean provides a baseline for which deviations are measured. It plays a pivotal role when predicting sums and averages using the Central Limit Theorem, allowing us to standardize distributions around this central value for further analysis.

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