Text: Questions 13 through 17: Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 146 "noted writers of the present" and are given in the following table. Mon Tues Wed Thurs Fri Sat Sun 17 26 22 23 18 15 25 13. Let Taay be the probability that a person is born on a particular day. What is the null hypothesis? a. Ho: TMon = TTue = TIWed = TThur = TIFri = Tsat = TSun 14. Fill in the table below with the expected counts based on the hypothesized model that people are equally likely to be born on any day of the seven days of the week. Mon Tues Wed Thurs Fri Sat Sun
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Information for questions 7 – 9: Claire can supposedly guess the month that a person was born in just by looking at them. You test this ability in the following way: you take 90 people to Claire and record the number whose birth-month she correctly "infers". Claire correctly guesses 11 of their birth-months. For simplicity, assume that a person has an equal chance of being born in any month, so that the probability of correctly guessing someone's birth month by pure chance is 1/12. 7. Suppose a person randomly guessed the birth-month of these 90 people, and that we called the proportion of correct guesses p̂. What is the (approximate) distribution of p̂? a) N(1/12, 1/12) b) N(1/12, 0.02913) c) N(12, 0.0465) d) N(0, 1) 8. Use the distribution from question 7 to find the probability that a person randomly guessing ninety people's birth-months would get at least seven right (i.e. that p̂ would be at least 0.0778). a) 0.7846 b) 0.2657 c) 0 d) 0.5756 9. Let pC be the proportion of people whose birth-month Claire correctly guesses; Claire's claim is equivalent to claiming that pC > 1/12. What are the p-value and conclusion of a hypothesis test of this claim at α = 0.05? a) p-value: 0.091, accept H0 b) p-value: 0.909, accept H0 c) p-value: 0.909, reject H0 d) p-value: 0.091, reject H0
Adi S.
Joshua A.
The article "Three Sisters Give Birth on the Same Day" (Chance, Spring $2001 : 23-25$ ) used the fact that three Utah sisters had all given birth on March $11,1998,$ as a basis for posing some interesting questions regarding birth coincidences. (a) Disregarding leap year and assuming that the other 365 days are equally likely, what is the probability that three randomly selected births all occur on March 11$?$ Be sure to indicate what, if any, extra assumptions you are making. (b) With the assumptions used in part (a), what is the probability that three randomly selected births all occur on the same day? (c) The author suggested that, based on extensive data, the length of gestation (time between conception and birth) could be modeled as having a normal distribution with mean value 280 days and standard deviation 19.88 days. The due dates for the three Utah sisters were March $15,$ April 1 , and April $4,$ respectively. Assuming that all three due dates are at the mean of the distribution, what is the probability that all births occurred on March 11$?$ $[$ Hint: The deviation of birth date from due date is normally distributed with mean $0 .]$ (d) Explain how you would use the information in part (c) to calculate the probability of a common birth date.
Continuous Random Variables and Probability Distributions
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