Probability & Statistics - Solutions

Transcription

1 Probability & Statistics - Solutions Question 1 (a) A survey of 50 students with part-time jobs, randomly selected in NUI Maynooth in January 2018, found that they had a mean weekly net pay of 374. The standard deviation of this sample was 45. Find the 95% confidence interval for the mean net pay of college students in NUI Maynooth. Interpret this interval. (b) The mean net weekly pay for all 1st year college students in Ireland, in 2017, was 385 and the standard deviation was 45. Aoife suggests that the mean income in NUI Maynooth is different from the whole country. Test this hypothesis using a 5% level of significance. Clearly state your null hypothesis, your alternative hypothesis and your conclusion.

2 Question 2 A sports scientist claims that the average distance walked each day by residents living in the town of Quahog is 3.5 km. To test this claim, a sample of 60 residents living in Quahog was randomly selected. The residents were asked to wear a pedometer for a week. The mean distance walked daily by the residents is 3.7 km with a standard deviation of 0.5 km. (a) Test the sports scientist s claim using a 5% level of significance. Clearly state your null hypothesis, your alternative hypothesis and your conclusion. (b) In the above sample of 60 residents, 20% of them walked 2.5 km daily. Find the 95% confidence interval for the proportion of residents who walk 2.5 km daily. (c) Data from 10 years ago shows that, at that time, 26% of the residents of Quahog walked 2.5 km daily. Based on your answer to part (b) is it possible to conclude, at the 5% level of significance, that the proportion of residents of Quahog who walked 2.5 km daily has changed since that time? Explain your answer.

3 Question 3 It is known that the results of a certain exam have a mean score of 70 and a standard deviation of 6. A class of 36 students are given this exam and they receive the mean result of (a) Calculate the sample statistic for this sample. (b) Calculate the p-value for this sample statistic. (c) Use the p-value to determine if the mean score of the sample differs from that of the population, at 5% significance.

4 Question 4 (a) Sweet Valley High has a very good basketball team. Liam is the best player. He is 208 cm tall. He takes most of his shots close to the basket and because he is so tall he scores 58% of his shots, each of which is worth 2 points. When Liam is fouled he gets two free shots, approximately 3 metres away from the basket, called foul shots. He only scores 51% of these shots which are worth 1 point each. The coach is concerned that opposing teams seem to be fouling Liam whenever he touches the ball, and he asks you to work out if this could pay off for them in the long run in terms of reducing Liam s scores. (i) Calculate the expected value of the number of points Liam scores on one regular shot at the basket. (ii) Calculate the expected value of the number of points Liam scores when he shoots two foul shots. Assume that all foul shots are independent events.

5 (b) You and a friend are playing the following game: Two dice are rolled. If the total showing is a prime number, you pay your friend 6, otherwise, your friend pays you 2. (i) What is the expected value of the game to you? (ii) If you played the game 40 times, what are your expected winnings? After playing the game for a while, you begin to think the rules are not fair and you decide to change the game. (iii) How much should you pay your friend (instead of 6) when you lose so that your expected winnings are exactly 0?

6 Question 5 (25 marks) (a) In a quiz, students are asked to solve two logical puzzles. 45% of the students solved the first problem. 25% of the students solved both problems. What is the probability that a student solved the second problem, given that the student had already solved the first problem? P(1) 0 45 P(1 2) 0 25 P(1 2) P(2 1) P(1) (b) The probability of a water pump being faulty is A random sample of 12 pumps is selected off the production line. Calculate the probability that: (i) only 1 pump is faulty, correct to two decimal places C (ii) no more than 3 pumps are faulty, correct to three decimal places C C C C