Question

Q1(a) Mr. Sweetheart likes sugar in his hot tea. He buys sugar packets from a local grocery store. Suppose the amount of sugar in a packet follows a normal distribution with mean 2.17 grams and standard deviation 0.08 grams. The amount of sugar in a package should be between 2.13 and 2.21 grams. Otherwise, a packet is considered to be defective.

i. What is the probability that a randomly selected packet is defective?

ii. Three sugar packets are chosen randomly. What is the probability that two of them are not defective?

iii. Mr Sweetheart needs between 8.2 and 9 grams of sugar in a cup of tea for the drink to taste right. One morning, he adds four randomly selected sugar packets. What is the probability that Mr Sweetheart’s tea tastes right?

Q1(b) President Fisher posts 2 messages on a social media platform daily on average.

i. What is the probability that he posts no message in three consecutive days?

ii. What is the probability that he posts more than one message in one hour?

Q2

A restaurant chain owner would like to know the sales of a certain signature dish in his restaurants. A random sample of 120 restaurants is selected and the sales of the dish in 2018 are summarized as follows.

Sales (in thousand ($))	Frequency
26 – 30	6
31 – 35	12
36 – 40	15
41 – 45	36
46 – 50	21
51 – 55	20
56 – 60	10

Q2(a) Calculate the mean, median and standard deviation of the sales. $)) Frequency 26 – 30 6 31 – 35 12 36 – 40 15 41 – 45 36 46 – 50 21 51 – 55 20 56 – 60 10

Q2(b) Calculate the coefficient of skewness using results in (a) and interpret your result briefly.

Q2(c) Estimate, from the frequency distribution table, the number of restaurants who sales were between $38,500 and $54,500 in 2018.

Q2(d) Estimate, from the frequency distribution table, the sales amount that exceeded by 30% of the restaurants in 2018.

Q3(a) To estimate the mean amount of time children spend in physical activities daily, a researcher randomly selects 8 children and records the number of hours they spend in physical activities in one day. The numbers of hours are: 4.1, 1.2, 4.6, 2.4, 3.2, 1.8, 2.4, and 0.7. Obtain a 95% confidence interval of the mean number of hours that children spend in physical activities.

Q3(b) It is found that the average lifespan of 100 smart phones is 23.5 months with a standard deviation of 10.2 months. Construct the 99% confidence interval for the mean lifespan of all smart phones. State the assumption(s) made.

Q3(c) A machine produces DVD discs. The diameters of these DVD discs vary, and the standard deviation is 0.01 centimeter. How large a sample should be taken if we wish to have 95% confidence that our sample mean will not differ from the true mean by more than 0.001 centimeter?

Q4 (a) On a sunny day, a theme park had 1,000 visitors. According to the attendance record, 800 visitors took a ride on the roller coaster; 450 visitors took a ride on the merry-go-round. It is estimated that among those visitors who took a ride on the roller coaster, 40% of them also took a ride on the merry-go-round. A visitor on that day is selected at random.

i. What is the probability that this visitor rode on both rides?

ii. What is the probability that this visitor rode on no rides at all?

iii. If this visitor has taken a ride on the merry-go-round, what is the probability that he has not ridden on the roller coaster?

Q4(b) In a tutorial session of AMA1501, there are 11 accounting students, 6 marketing students and 8 financial service students. Among these 25 students, a group of 5 students is selected randomly for the first presentation.

i. How many different groups can be formed?

ii. What is the probability that this group consists of only accounting students?

iii. What is the probability that this group consists of exactly 2 accounting students and 3 marketing students?

Q4(c) Three urns contain colored balls. Urn 1 contains 3 red, 4 white and 1 blue balls. Urn 2 contains 4 red, 3 white and 2 blue balls. Urn 3 contains 1 red, 2 white and 3 blue balls. One urn is chosen at random and a ball is drawn from it. If the ball is red, what is the probability that it came from Urn 3?

Answer 1

1(i)

By the complement rule, the probability that a randomly selected packet is defective is

1 - 0.383 = 0.617

Answer: 0.617

(ii)

Here we need to use binomial distribution with parameters n=3 and p=0.383 (from part a). The probability that two of them are not defective is

(iii)

Let T shows the total amount of sugar in four packets. So distribution of T will be approximately normal with mean and SD as follows:

The z-score for T = 8.2 is

The z-score for T = 9 is

The probability that Mr Sweetheart’s tea tastes right is

Q1(b)(i)

Let X is a random variable shows the number of messages in 3 days. Here X has Poisson distribution with parameter . The probability that he posts no message in three consecutive days is

ii. Since 1 day =24 hours so average number of messages per hour is

The probability that he posts more than one message in one hour is