Semhar works as a waiter in a local pizza restaurant where he waits on about 40

Question

Semhar works as a waiter in a local pizza restaurant where he waits on about 40 parties over a weekend of work and 10 parties each day he works during the week. After collecting some careful data on his tips over the last year, he believes the distribution of his tips has a model taht 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?

d) Estimate the probability that he will earn at least $420 in tips over a weekend.

e) How much does he earn on the best 10% of weekends?

Explanation / Answer

a) Since the distribution of tips is skewed to the right, we cant use the Normal model to determine the probability that a given party will tip at least $20

b) No. A sample of 4 parties is probably not a large enough sample for the CLT to allow us to use the Normal model to estimate the distribution of averages.

c) A sample of 10 parties may not be large enough to allow the use of a Normal model to describe the distribution of averages. It would be risky to attempt to estimate the probability that his next 10 parties tip an average of $15. However, since the distribution of tips has µx = $9.60 , with standard deviation x = $5.40, we still know that the mean of the sampling distribution model is µx¯ = $9.60 with standard deviation SD

x = 5..40/ 10 ~ 1.71

d)  We’re assuming that these 40 parties can be considered as a random sample of all tips and that the tips from each party are independent of each other.

Recall that you should have under 10% of all tips (in order to maintain that independence) and a sample of size 40 seems large enough. In order to earn $420 in a weekend, the average tip from a party needs to be $420/40=$10.50.

Recall µx = 9.60 and x = 5.40. Since the above conditions are satisfied, we can use the central limit theorem to model the sampling distribution of the average tip with µx¯ = 9.60 and x¯=x = 5..40/ 40 ~ 0.8538
z =10.5 -9.6 / 0.8538 ~ 1.05

e) We would want to leave 10% of area in the right tail of the distribution, so look up a z value with a

CDF of 0.9; z=1.2816.

1.2816 = x9.6 /0.8538

x = 10.69.

The average tip on the best 10% of weekends is about $10.69, so multiply that by 40 parties and you get about $427.77 for the weekend.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.